Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich 7 Philippe Tondeur Department of Mathematics University of ZfJrich Introduction to Lie Groups and 1965 Transformation Groups Springer-Verlag. Berlin-Heidelberg. New York
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Introduction to Lie Groups and Transformation Groups
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Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics
Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
7
Philippe Tondeur Department of Mathematics
University of ZfJrich
Introduction to Lie Groups
and
1965 Transformation Groups
Springer-Verlag. Berlin-Heidelberg. New York
All rights, especially that oftranalation/nto foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without
written permission from Springer Verlag. @ by Sprlnger-Verlag Berlin �9 Heidelberg 196~. Library of Congress Catalog Card Number 6~--26947. Printed in Germany. Title No. 7327
Printed by Behz, Weinhelm
PREFACE
T h e s e n o t e s w e r e w r i t t e n f o r i n t r o d u c t o r y l e c t u r e s on L i e g r o u p s
a n d t r a n s f o r m a t i o n g r o u p s , h e l d a t t h e U n i v e r s i t i e s of B u e n o s A i r e s
a n d Z u r i c h . T h e n o t i o n s of a d i f f e r e n t i a b l e m a n i f o l d , a d i f f e r e n t i a b l e
m a p a n d a v e c t o r f i e l d a r e s u p p o s e d k n o w n . T h e r e i s a n a p p e n d i x on
c a t e g o r i e s a n d f u n c t o r s .
T h e f i r s t t w o c h a p t e r s a r e i n f l u e n c e d b y a p a p e r of R . P a l a i s [ l g ] .
In s e c t i o n s 5. Z a n d 5. 3, a l o t i s t a k e n f r o m S. K o b a y a s h i a n d K. N o m i z u
[11] . In c h a p t e r 7, S. H e l g a s o n [61 w a s o f t e n u s e d . Of c o u r s e ,
C. C h e v a l l e y [ 3] w a s c o n s t a n t l y c o n s u l t e d . T h e b i b l i o g r a p h y o r i e n t s
on t h e v a r i o u s s o u r c e s . A s p e c i a l f e a t u r e of t h i s p r e s e n t a t i o n i s t h e
s y s t e m a t i c a v o i d a n c e of t h e u s e of l o c a l c o o r d i n a t e s on a m a n i f o l d . T h i s
a l l o w s t h e u s e of t h e p r e s e n t e d t h e o r y w i t h s l i g h t m o d i f i c a t i o n s f o r L i e
g r o u p s o v e r B a n a c h m a n i f o l d s . S e e e . g . B . M a i s s e n [10].
J u n e 1964 P h i l i p p e T o n d e u r
CONTENTS
.
.
.
.
G - O b j e c t s .
1.1.
1. Z.
1 .3.
" 1 . 4 .
D e f i n i t i o n a n d e x a m p l e s .
E q u i v a r i a n t m o r p h i s m s .
O r b i t s .
P a r t i c u l a r G - s e t s .
G - S p a c e s .
Z. 1. D e f i n i t i o n a n d e x a m p l e s .
Z .Z. O r b i t s p a c e .
G - M a n i f o l d s .
3.1. D e f i n i t i o n a n d e x a m p l e s of L i e g r o u p s .
3. Z. D e f i n i t i o n and e x a m p l e s of G - m a n i f o l d s .
V e c t o r f i e l d s .
4. 1. R e a l f u n c t i o n s .
4. Z. O p e r a t o r s a n d v e c t o r f i e l d s .
4 . 3 . T h e L i e a l g e b r a of a L i e g r o u p .
4 . 4 . E f f e c t of m a p s on o p e r a t o r s and v e c t o r f i e l d s .
4. 5. T h e f u n c t o r L.
4. 6. A p p l i c a t i o n s of t h e f u n c t o r a l i t y of L.
4. 7. The adjoint representation of a Lie group.
1
7
13
23
28
30
34
37
40
4Z
46
50
5Z
59
64
The * indicates a section, the lecture of which is not necessary for the
understanding of the subsequent developments.
.
.
.
V e c t o r f i e l d s a n d 1 - p a r a m e t e r ~ r o u p s of t r a n s f o r m a t i o n s .
5.1.
5. Z.
5. 3.
5 . 4 .
5 . 5 .
* 5 . 6 .
*5. 7.
1 - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s .
1 - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s a n d e q u i v a r i a n t m a p s .
T h e b r a c k e t of t w o v e c t o r f i e l d s .
1 - p a r a m e t e r s u b g r o u p s of a L i e g r o u p .
K i l l i n g v e c t o r f i e l d s .
T h e h o m o m o r p h i s m aV: RG > DX f o r a G - m a n i f o l d .
K i l l i n g v e c t o r f i e l d s a n d e q u i v a r i a n t m a p s .
T h e e x p o n e n t i a l m a p of a L i e g r o u p .
6.1.
6. Z.
6.3.
*6.4.
6.5.
D e f i n i t i o n a n d n a t u r a l i t y of e x p .
e x p is a l o c a l d i f f e o m o r p h i s m at t h e i d e n t i t y .
U n i c i t y of L i e g r o u p s t r u c t u r e .
A p p l i c a t i o n to f i x e d p o i n t s on G - m a n i f o l d s .
T a y l o r ' s f o r m u l a .
S u b g r o u p s a n d s u b a l g e b r a , s .
7 .1. L i e s u b g r o u p s .
7. Z. E x i s t e n c e of l o c a l h o m o m o r p h i s m s .
7. 3. D i s c r e t e s u b g r o u p s .
7 . 4 . O p e n s u b g r o u p s ; c o n n e c t e d n e s s .
7 . 5 . C l o s e d s u b g r o u p s .
7 . 6 . C l o s e d s u b g r o u p s of t h e f u l l l i n e a r g r o u p .
7. 7. Coset spaces and factor groups.
66
70
74
77
84
89
96
103
108
1 iZ
116
IZO
128
132
138
142
144
150
154
. G r o u p s of a u t o m o r p h i s m s .
8.1. The a u t o m o r p h i s m g roup of an a l g e b r a .
8. Z. The ad jo in t r e p r e s e n t a t i o n of a L i e a l g e b r a ,
8. 3. The a u t o m o r p h i s m g r o u p of a L i e g roup ,
160
16Z
167
Append ix : C a t e g o r i e s and f u n c t o r s . 170
B i b l i o g r a p h y 175
-1-
C h a p t e r l . G - O B J E C T S
The first two paragraphs of this chapter are essential for all that
follows, whereas paragraphs i. 3 and i. 4 are only required for the
lecture of g. g and shall not be used otherwise. For the notion of
category and functor, see appendix.
i. i Definition and examples.
If X is an object of a category ~ , we denote by Aut X the group
of equivalences of X with itself. Let G be a group.
DEFINITION I. i. I An operation of G on X is a homomorphisrn
r: G >Aut X. X is called a G-object with respect to T.
on X is a representation of G by automorphisrns An o p e r a t i o n of G
of X .
E x a m p l e 1 .1 .2
s e t X
of X .
by the s a m e l e t t e r )
A G - o b j e c t X in the c a t e g o r y of s e t s E n s is a
e q u i p p e d w i t h a h o m o m o r p h i s m 7 of G in to the g r o u p of b i j e c t i o n s
Such a homomorphism is equivalently defined by a map (denoted
G x X > X
(g, x) ~ - ~ ~ v (x) g
satisfying
-Z-
a) (x) = v ( v (x)) for gz C G xC X Tglgz gl gz gl' '
b) T (x) = x f o r e C G, x C X e
The last conditions in the example i. i. Z suggest calling an
operation in the sense of definition i. i. i more precisely a left-
o p e r a t i o n of G on X . A r i g h t - o p e r a t i o n of G on X w i l l t h e n
G ~ be a h o m o m o r p h i s m ~" : > A u t X, w h e r e G ~ i s t h e
o p p o s i t e g r o u p of G , i . e . t h e u n d e r l y i n g s e t of G w i t h t h e
m u l t i p l i c a t i o n (g lgg) o o = g z g 1. X i s t h e n a G - o b j e c t . We
s h a l l g e n e r a l l y u s e t h e w o r d o p e r a t i o n a s s y n o n y m o u s f o r
left-operation and only be more precise when right-operations
also occur.
Example 1.1. 3 Let G be a group. If to any g C G we
assign the corresponding left translation L of G defined by g
L (~/) = gv f o r V g G , g
we obtain a left-operation of G on
the underlying set of G. Similarly, the assignment of the right
translation Rg of G, Rg(V) = Vg for V C G, to any g 6 G
defines a right-operation of G on the underlying set of G.
Example 1. I. 4 Let p : G
of groups. It defines an operation
set of G' in the following way: set
> G' be a homomorphism
r of G on the underlying
= Lp(g) for g C G.
-3 -
One o b t a i n s s i m i l a r l y a r i g h t o p e r a t i o n
= Rp f o r g g G O-g (g) .
o- by the definition
E x a m p l e 1.1. 5 L e t G be a g r o u p . T o a n y g C G we
a s s i g n t h e i n n e r a u t o m o r p h i s m g
-1 induced by g, ~I (~) = g~/g
g
f o r ~ C G . T h i s d e f i n e s an o p e r a t i o n of G on i t s e l f .
E x a m p l e 1 .1 .6
c o n s i d e r the m a p G x H
m u l t i p l i c a t i o n O x O
H on the s e t O .
L e t H be a s u b g r o u p of t h e g r o u p G and
> G d e f i n e d by r e s t r i c t i n g the
> O . It d e f i n e s a r i g h t - o p e r a t i o n of
by
l a w
E x a m p l e 1.1. 7 L e t t h e g r o u p G
T : G > Au t G ' . On the s e t G'
o p e r a t e on t h e g r o u p G'
x G the m u l t i p l i c a t i o n
(gl" g l ) (gz" gz ) = (gl ''r gl (gz ')' glg2 )
f o r g i ' g G ' , g i g G (i = ~, Z)
d e f i n e s a g r o u p s t r u c t u r e , t h e s e m i - d i r e c t p r o d u c t d e n o t e d
O ' x v G . C o n s i d e r t he h o m o m o r p h i s m s
j : G ' > G ' x~: G j (g ' ) = (g ' , e) f o r g' 6 G ' , e n e u t r a l i n G
p : G' x ~ G > G p (g ' , g) = g f o r g' g G ' , g E G
s : G > G' x G s(g) = (e' g) for e' neutral in G' g GG T J
-4 -
T h e s e q u e n c e
(*) e > G' J > P P > G > e
with P = G'x G is exact and s satisfies p oS = 1 G Conversely T
an e x a c t s e q u e n c e (*) and a h o m o m o r p h i s m s : G > P w i t h
p o s = 1G (a s p l i t t i n g of (*)) d e f i n e s an o p e r a t i o n T of G on G v :
the a u t o m o r p h i s m -r o f G ' c o r r e s p o n d i n g to g ~ G i s the g
i n n e r a u t o m o r p h i s m of P d e f i n e d b y s ( g ) , r e s t r i c t e d to the
n o r m a l s u b g r o u p G ' . T h e r e f o r e G - g r o u p s a r e in ( 1 - 1 ) - c o r r e s p o n d e n c e
w i t h s p l i t t i n g e x a c t s e q u e n c e s (*).
E x a m p l e 1.1. 8 A t y p i c a l c a s e of the s i t u a t i o n j u s t m e n t i o n e d
i s a s f o l l o w s : L e t V be a f i n i t e - d i m e n s i o n a l I R - v e c t o r s p a c e
and G L ( V ) the g r o u p of l i n e a r a u t o m o r p h i s r n s of V. T h e n G L ( V )
o p e r a t e s n a t u r a l l y on V . T h e s e m i - d i r e c t p r o d u c t V x G L ( V )
i s the g r o u p of a f f i n e m o t i o n s of V . N o t e t h a t t he m u l t i p l i c a t i o n
j u s t c o r r e s p o n d s to the n a t u r a l c o m p o s i t i o n of a f f i n e m o t i o n s .
We s h a l l on ly h a v e to c o n s i d e r c a t e g o r i e s ~ w h o s e o b j e c t s
h a v e an u n d e r l y i n g s e t an d w h o s e m o r p h i s m s a r e a p p l i c a t i o n s
of the u n d e r l y i n g s e t s . M o r e p r e c i s e l y t h i s m e a n s t h a t t h e r e
e x i s t s a f u n c t o r V : ~ > E n s w h i c h c a n be t h o u g h t of a s
f o r g e t t i n g a b o u t the a d d i t i o n a l s t r u c t u r e on X in R and t a k i n g
a m o r p h i s m j u s t a s an a p p l i c a t i o n . T o a v o i d e n d l e s s r e p e t i t i o n s
- 5 -
we m a k e t h e following c o n v e n t i o n : F r o m n o w on we s h a l l
o n l y c o n s i d e r c a t e g o r i e s of t h a t s o r t a n d s h a l l u s e t h e s a m e
n o t a t i o n X f o r a n o b j e c t X a n d i t s u n d e r l y i n g s e t V X .
A n o p e r a t i o n of t h e g r o u p G on X d e f i n e s a n o p e r a t i o n on
t h e u n d e r l y i n g s e t . M o r e g e n e r a l l y w e h a v e t h e
P R O P O S I T I O N 1 . 1 . 9
f u n c t o r f r o m t h e c a_tegor_~y
of t h e group G
FX ~ ~.'
on X C~
Let F : R --> g' be a c ovariant
to the category R'. An operation
induces a well-defined operation on
Proof: F defines a homomorphism Aut X
By composition with the given homomorphism G
> Aut FX.
> Aut X we
functor F : ~. ISO
obtain a homomorphism G > Aut FX, which is the desired
operation of G on FX.
Remark I. I. I0 If in a given category ~ we only consider
equivalences as morphisms, we obtain a new category ~. Iso
Evidently proposition I. I. 9 is still valid if we are only given a
> ~I. ISO
If F :~ > ~' is a contravariant functor, a left-operation
of G on X induces a right-operation of G on FX, and a right-
operation on X is turned into a left-operation on FX.
Example 1.1.11 Consider the covariant functor
P : Ens > Ens, making correspond to each set X the set PX
- 6 -
of i t s s u b s e t s , to e a c h m a p X
P X > P X ' of s u b s e t s . L e t X be a G - s e t .
-1 G - s e t by p r o p o s i t i o n 1 .1 .9 . T h e f u n c t o r P
h a v i n g the s a m e e f f e c t a s P
> X' t h e i n d u c e d m a p
T h e n P X is a
: E n s > E n s ,
o n o b j e c t s of E n s , but a s s i g n i n g
-1 to a m a p @ : X > X ' t h e m a p r : P X ' > P X ( i n v e r s e
images of subsets), transforms the G-set X into the G ~ -set
P X .
E x a m p l e 1.1.12
T h e c o n t r a v a r i a n t f u n c t o r
L e t R be a f i x e d o b j e c t of t h e c a t e g o r y
hR(X) = [ X , R ] , = f o
g i v e s f o r a n y l e f t - o p e r a t i o n of G
on the s e t [ X , R] . If T : G
p h i s m , we w r i t e T
i n t o t h e g r o u p of b i j e c t i o n s o f [ X , R ] .
E x a m p l e 1 . 1 . 1 3 L e t A be a r i n g ,
hR : ~ > E n s d e f i n e d by
f o r f E ; [ X t, R] , ~ : X > X ' ,
on X a r i g h t - o p e r a t i o n of G
> Au t X is the g i v e n h o m o m o r -
f o r t h e i n d u c e d h o m o m o r p h i s m of G ~
2l the c a t e g o r y of
l e f t - A - m o d u l e s . A G - m o d u l e X i s d e f i n e d by a n o p e r a t i o n of
G on X by A - l i n e a r m a p s ; i . e . a r e p r e s e n t a t i o n of G in X
in t he u s u a l s e n s e . By p r o p o s i t i o n 1 .1 .9 s u c h a r e p r e s e n t a t i o n
i n d u c e s an o p e r a t i o n of G on the s e t of s u b m o d u l e s of X .
F o l l o w i n g o u r c o n v e n t i o n on t h e c a t e g o r i e s to c o n s i d e r , i t
m a k e s s e n s e to s p e a k of an e l e m e n t of an o b j e c t X .
-7-
D E F I N I T I O N i. i . 14 A n e l e m e n t x i n t h e G - o b j e c t
c a l l e d i n v a r i a n t o r G - i n v a r i a n t i f x i s f i x e d u n d e r e v e r y
transformation m g
A subset M c
X is
: Tg(X) = x f o r a l l g ~ G .
X i s c a l l e d i n v a r i a n t i f i t i s a n i n v a r i a n t
element of PX under the induced G-operation (example i. I. Ii),
i . e . i f T (M) c M f o r a l l g C G . g
E x e r c i s e 1 . 1 . 1 5 L e t X a n d X ' b e G - o b j e c t s of ~ w i t h
respect to T: G >Aut X and I-' : G >Aut X'.
O-g(~) = T' g o ~ o ~" -I for g G G, ~: X >X' defines an g
operation of G on the set of morphisms from X to X'.
(Example i. 1. IZ is a special case of this situation, if we considert
trivial G-operation on X'.) Show that there is a suitable functor
inducing this operation according to proposition I. I. 9.
1 . 2 E q u i v a r i a n t m o r p h i s m s .
L e t G a n d G ' b e g r o u p s a n d R a c a t e g o r y . S u p p o s e X
w i t h r e s p e c t t o a h o m o m o r p h i s m
w i t h r e s p e c t t o a h o m o m o r p h i s m
a G ' - o b j e c t of ~ w i t h r e s p e c t t o a
O' > Aut X'.
A p-equivariant morphism
P:G
t o b e a G - o b j e c t of
T: G > A u t X , X '
! h o m o m o r p h i s m v :
D E F I N I T I O N I. 2. i
~: X --> X' > G '
- 8 -
i s a m o r p h i s m ~0: X > X'
f o l l o w i n g d i a g r a m c o m m u t e s
of g s u c h t h a t f o r a l l g ~ G t h e
T g
X ~o > X'
X
I
d T
>X
map.
If G =G' and P = I G, we just speak of anequivariant
E x a m p l e 1. Z. Z If X i s a G - s e t a n d X' a G ' - s e t w i t h t h e
o p e r a t i o n s g i v e n a s i n e x a m p l e 1.1. ~, t h e n a m a p ~ : X > X'
i s P - e q u i v a r i a n t i f a n d o n l y i f t h e f o l l o w i n g d i a g r a m c o m m u t e s
GxX T > X
I ' d , T
G ' x X t . > X t
E x a m p l e 1. Z. 3 L e t P : G - - > G v be a h o m o m o r p h i s m
of g r o u p s . I f G a n d G ' a r e o p e r a t i n g on i t s e l f by l e f t -
t r a n s l a t i o n a s i n e x a m p l e 1.1. 3, t h e n a m a p q~ : G > G ' i s
p - e q u i v a r i a n t i f a n d o n l y i f ~ ( g l g 2 ) = p ( g l ) ~ ( g z ) . T h e r e -
f o r e p i t s e l f i s a n e x a m p l e of a p - e q u i v a r i a n t m a p w i t h
r e s p e c t to t h e l e f t - o p e r a t i o n s .
- 9 -
If w e c o n s i d e r t h e o p e r a t i o n s of G a n d G '
i n n e r a u t o m o r p h i s m s ,
on i t s e l f b y
commutes, i.e. p is
Example I. Z. 4
t h e n f o r a l l g C G t h e d i a g r a m
G P > G'
I I
G P > G'
p - e q u i v a r i a n t .
If w e c o n s i d e r t h e r i g h t - o p e r a t i o n of t h e
s u b g r o u p H of G on G a s i n e x a m p l e 1 . 1 . 6 , t h e n a h o m o m o r p h i s m
P : G - - > G s e n d i n g H i n t o H c a n b e c o n s i d e r e d a s a p / H -
p / H d e n o t e s t h e r e s t r i c t i o n of p e q u i v a r i a n t m a p , w h e r e
t o H .
E x a m p l e 1. Z. 5 A n y r i g h t - t r a n s l a t i o n of a g r o u p G i s a n
e q u i v a r i a n t m a p of t h e G - s e t G d e f i n e d b y t h e l e f t - t r a n s l a t i o n .
T h i s i s j u s t t h e a s s o c i a t i v i t y l a w i n G .
Example i. Z. 6 If
of G on X, then for any g ~ G the map
- equivariant, where g
automorphism of G defined by g.
Example i. Z. 7 Let X be a G-set.
p ( g ) = "r
T: G > A u t X d e f i n e s a n o p e r a t i o n
T :X >X i s g
[~ : G - - > G d e n o t e s t h e i n n e r g
g(X o) de fines a map p : G
F o r f i x e d x C X o
> X . I f w e c o n s i d e r t h e
- 1 0 -
o p e r a t i o n of G on G by l e f t - t r a n s l a t i o n , p is an e q u i v a r i a n t
map.
If X, X', X" are G, G,
9:G >G', P' :G'
~0: X >X', ~01 : X'
respectively, then clearly
G"-objects respectively,
> G" homornorphisms and
> X" p, p'-equivariant morphisms
! !
~0 o r is a P o p-equivariant
m o r p h i s m . F o r f i x e d G the G - o b j e c t s of a c a t e g o r y ~ t h e r e -
G f o r e f o r m a c a t e g o r y ~ w i t h t h e e q u i v a r i a n t m o r p h i s m s as
m o r p h i s m s ( D e f i n i t i o n 1 . 2 . 8 ) .
As a c o m p l e m e n t to p r o p o s i t i o n 1 .1 .9 we h a v e
P R O P O S I T ION 1. Z. 9 L e t F : K > K' b e a c o v a r i a n t
functor, X, X' respectively G, G'-objects of ~ , p : G~> G'
a h o m o m o r p h i s m and q~: X > X' a p-equivariant morphism.
!
C o n s i d e r the n a t u r a l o p e r a t i o n s i n d u c e d on F X and F X T h e n
F ( ~ ) : F X > F X ' is P - e q u i v a r i a n t w i t h r e s p e c t to t h e s e
o p e r a t i o n s . F o r a f i x e d g r o u p G th i s d e f i n e s in p a r t i c u l a r a n
extension of the map of proposition 1.1.9, sending G-objects of
into G-objects of K' , to a functor F G ~G ~,G : > .
-ii-
Proof: The commutative diagram
X ~ ,> X' r
% r [
~o X ~ X -- ' >
f
i s t r a n s f o r m e d by F in t h e c o m m u t a t i v e d i a g r a m
FX F(~) > FX'
r T
F(Tg) i i F ( ' ~
FX F(~) > FX'
p(g)
s h o w i n g t h e P - e q u i v a r i a n c e of F(~) w i t h r e s p e c t to t h e
i n d u c e d o p e r a t i o n on F X a n d F X ' . T h e r e s t i s c l e a r .
If an e q u i v a r i a n t m o r p h i s r n
!
i n v e r s e ~ : X > X in R , i . e .
t h e n ~ i s n e c e s s a r i l y e q u i v a r i a n t ,
G e q u i v a l e n c e in ~ .
T h e r e i s a c a n o n i c a l f u n c t o r V : R G
c o n s i s t s i n f o r g e t t i n g a b o u t t h e G - o p e r a t i o n .
~0 : X > X ' in ~G h a s an
% o ~0 = l x , ~0 o ~ = I x , ,
and ~0 t h e r e f o r e an
> ~ w h i c h
On t h e o t h e r h a n d ,
-12 -
w e d e f i n e a f u n c t o r I : ~ > ~G b y c o n s i d e r i n g on e v e r y
o b j e c t X of ~ t h e t r i v i a l G - o p e r a t i o n 7: G > A u t X
m a p p i n g G on 1 x . T h e r e f o r e i t m a k e s s e n s e t o s p e a k of
e q u i v a r i a n t m o r p h i s m s ~0: X > R , w h e r e X i s a G - o b j e c t
of ~ a n d R a n a r b i t r a r y o b j e c t of e . W e c a l l s u c h a m a p
a n i n v a r i a n t m o r p h i s m . M o r e p r e c i s e l y w e h a v e t h e
D E F I N I T I O N 1 .2 .11 L e t X be a G - o b j e c t of ~ , R a n
o b j e c t of ~ . A m o r p h i s m ~0 : X > R in ~ i s c a l l e d
i n v a r ! a n t i f f o r a l l g G G t h e f o l l o w i n g d i a g r a m c o m m u t e s
X
X
P R O P O S I T I O N 1. Z. 12 L e t X b e a G - s e t , X ' I
a G - s e t
a n d ~9 : X > X ' a 9 - e q u i v a r i a n t m a p w i t h r e s p e c t t o _a
h o m o r n o r p h i s m 9 : G > G ' . If x 6 X i s G - i n v a r i a n t , t h e n
(x) i s p ( G ) - i n v a r i a n t .
P r o o f : !
Tg(X) = x i m p l i e s T p ( g ) ( ~ ( x ) ) = ~ ( T g ( X ) ) = ~ ( x ) .
-13-
A s a c o n s e q u e n c e j G - i n v a r i a n t s u b s e t s of X go i n t o
p ( G ) - i n v a r i a n t s u b s e t s of X t .
E x e r c i s e 1 . 2 . 1 3 ~G c a n be c o n s i d e r e d a s a c a t e g o r y of
f u n c t o r s ( i n t e r p r e t G as a c a t e g o r y c o n s i s t i n g of a s i n g l e o b j e c t
and m o r p h i s m s g w i t h g C G w i t h n a t u r a l c o m p o s i t i o n l aw) .
E q u i v a r i a n t m o r p h i s m s a r e t h e n j u s t n a t u r a l t r a n s f o r m a t i o n s .
T h e f u n c t o r F G of p r o p o s i t i o n 1 . 2 . 9 is t he c a n o n i c a l f u n c t o r
i n d u c e d b y F b e t w e e n the c o r r e s p o n d i n g f u n c t o r c a t e g o r i e s .
E x e r c i s e 1. Z .14 If X and X' a r e G - o b j e c t s of ~ , t h e
s e t of m o r p h i s m s f r o m X to X' i s a G - s e t a c c o r d i n g to
e x e r c i s e 1 .1 .15 . T h e i n v a r i a n t e l e m e n t s u n d e r t h i s o p e r a t i o n
a r e the e q u i v a r i a n t m o r p h i s m s X - - > X ' . As a s p e c i a l c a s e ,
t he i n v a r i a n t m o r p h i s m s X > R , w h e r e R i s an o b j e c t of
, a r e t h e i n v a r i a n t e l e m e n t s u n d e r the o p e r a t i o n d e f i n e d in
e x a m p l e 1.1.1Z.
1 . 3
the g i v e n o p e r a t i o n i s the s e t
O r b i t s .
L e t X b e a G - s e t w i t h r e s p e c t t o T: G - > A u t X.
D E F I N I T I O N 1. 3.1 T h e o r b i t o r G - o r b i t of x C X u n d e r
~(x) = [ "rg(X)/g. G G } .
- 1 4 -
L E M M A I. 3.2. If
o r b i t s f o r m a p a r t i t i o n of
X i s a G - s e t , t h e d i f f e r e n t
X i n t o d i s j o i n t s e t s .
P r o o f : A s x g ~(x) , t he o r b i t s c o v e r X . We
o n l y h a v e to s h o w : i f two p o i n t s x , x ' g X h a v e i n t e r -
s e c t i n g o r b i t s [2(x), ~(x ' ) , t h e n ~(x) = ~(x ' ) . L e t
y ~ ~2(x) (] ~2(x' ): y = Vg(X), y = Vg, (x') . For z ~ ~2(x):z
= "rX(x ) we h a v e z = (-rye -rg_ 1 . - r g , ) ( x ' ) ~ ~ { x ' ) , i . e .
~(x) c ~(x' ), This shows ~(x) = ~2(x' ) .
Let X/G be the set of orbits, ~x:X -. X/G the
canonical map. An orbit is the orbit of any of its points.
This implies ~(Vg(X)) = =(x), i.e. ~ is an invariant map.
More generally we have
L E M M A 1 . 3 . 3 . L e t X be a G - s e t , m , ,
t he c a n o n i c a l m a p on to i t s o r b i t s e t X / G a n d
a r b i t r a r y , s e t . F o r a n y i n v a r i a n t m a p r X -. R
i s o n e a n d o n l y one m a p 4 : X / G -. R s u c h t h a t
~X: X -- X / G
R a n
t h e r e
= ~,Ir x
P r o o f : I f ~ , "rg = r f o r a l l g ~: G , t h e n
i s c o n s t a n t on e a c h o r b i t ~(x) , a n d t h e r e f o r e d e f i n e s a
m a p @: X / G -. R w i t h t h e d e s i r e d p r o p e r t y .
-15 -
On t h e o t h e r h a n d , a m a p x~ : X / G
Tr x : X > X / G g i v e s a n i n v a r i a n t m a p
h a v e p r o v e d
> R c o m p o s e d w i t h
~o= ~ o ~ W e X"
P R O P O S I T I O N 1. 3 . 4 L e t X be a G - s e t , WX : X > X / G
t h e c a n o n i c a l m a p o n t o i t s s e t of o r b i t s a n d R a n a r b i t r a r y s e t .
T h e c o r r e s p o n d e n c e % ~ - ~ - > % o ~ , s e n d i n ~ m a p s f r o m X / G
t o R i n t o i n v a r i a n t m a p s f r o m X t_~o R i s b i j e c t i v e .
Remark. X/G is characterized by this universal
property up to a canonical bijection by a standard argument.
This property allows therefore the definition of X/G in an
arbitrary category. Of course, there remains to show the
existence of such an orbit-object in a given category.
PROPOSITION I. 3.5 Let X be a G-set, X' a G'-set,
p : G >G' a homomorphism and q~ : X >X' a
9 - e q u i v a r i a n t m a p . T h e n t h e r e e x i s t s o n e a n d o n l y one m a p
~: X/G > X'/G' , such that the following diagram commutes
Q0 X >X
I I
~ x ~ X / G ' > / G t
-16 -
P r o o f : By t h e u n i v e r s a l p r o p e r t y s t a t e d in p r o p o s i t i o n 1. 3 . 4 ,
it is sufficient to show that w x, o ~ : X > X t/G t is an invariant
map. But
( ~ x ' ~ ~ ) ~ Tg = = x ' ~ ( ~ ~ *g) = ~ x ' o(T;(g) - 0 )
!
= (~r x, o Tp(g ) ) o ~ = w x, o ~2 WXV b e i n g
an i n v a r i a n t m a p . ~ is n o w d e f i n e d as t h e f a c t o r i z a t i o n of
,r x, o ~ through X / G .
E x a m p l e 1. 3. 6 L e t G be a g r o u p a n d H a s u b g r o u p ,
o p e r a t i n g by r i g h t t r a n s l a t i o n s on G ( e x a m p l e 1. 1 .6) . T h e n G / H
d e n o t e s t h e s e t of o r b i t s , t he s e t of l e f t c o s e t s m o d u l o H . L e t
G v be a n o t h e r g r o u p and H v a s u b g r o u p of G v L e t f u r t h e r
: G > G v be a m a p s u c h t h a t ~ ( H ) C H v a n d ~ ( g h )
= ~0(g}~0(h) f o r g ~ G , h ~ H . T h e n ~o/H : H > H' i s a
h o m o m o r p h i s m and ~ is ~ / H - e q u i v a r i a n t . B y p r o p o s i t i o n
1. 3 .5 t h e r e e x i s t s one a n d o n l y one m a p ~0 : G / H > GV/H t
s u c h t h a t t h e d i a g r a m
G ~ > G I
r l Gf f - G ,
G / H > G /H '
-17-
c o m m u t e s . In t h e c a s e w h e r e H and H' a r e n o r m a l s u b g r o u p s
of G and G' r e s p e c t i v e l y and ~0 i s a h o m o m o r p h i s m , (p is t he
i n d u c e d h o m o m o r p h i s m of the q u o t i e n t g r o u p s .
C o n s i d e r n o w a f i x e d g r o u p G . F o r a n y G - s e t X we h a v e
d e f i n e d the o r b i t s e t X / G . M o r e o v e r by p r o p o s i t i o n 1. 3 .5 a n y
e q u i v a r i a n t m a p (p: X - > X' i n d u c e s one and o n l y one m a p
r X / G > X ' / G . In t h i s w a y we o b t a i n a c o v a r i a n t f u n c t o r
Ens G "~ B : ~ Ens from G-sets to sets: B(X) = X/G, B(~0) : (p.
A standard consequence is that an equivalence ~0: X > X'
in Ens G induces a bijection ~0: X/G .... > X'/G.
Remark. If we consider the "forget-functor" V: Ens G
defined as forgetting about the G-set structure, we see that
w : V >B is a natural transformation of V into B.
To the beginning of this paragraphjfor a G-set X 2 we have
introduced the map
by the map w x : X
can be extended to a map PX --> PX and we interpret now
-i as the map w x , w x : PX > PX. For M c X Q(M) is
just the orbit of M under the induced G-operation on PX.
Explicitly
> Ens
: X > P X , w h i c h c a n a l s o be d e s c r i b e d
-1 X / G as ~ = ~ X ~ T h e r i g h t s i d e
- (-,- (x) l g c G , x e ; M } . g
-18-
2 ( M ) i s t h e r e f o r e t h e s a t u r a t i o n of M w i t h r e s p e c t t o G ,
i . e . t h e u n i o n of a l l G - o r b i t s of X i n t e r s e c t i n g M .
T h e i n v a r i a n c e of M c X c a n n o w be e x p r e s s e d by
(M) = M. For an arbitrary M c X the set
intersection of all invariant sets containing M.
are the minimal invariant sets.
~ ( M ) i s t h e
T h e o r b i t s
CfK-"
Isotropy groups.
{g e G / = x }.
Let X be a G-set and x C X.
G is a subgroup of G. x
Consider
DEFINITION i. 3.7. G is called the isotropy group of x. x
= Gxg-I PROPOSITI ON I. 3.8. Gg x g
Proof:
h6G x
g Gxg -1 c G gx
g-1 c G G g x g x
For simplicity we write Vg(X) = gx. Then for
we have ghg -I . gx = ghx = gx, which implies
A s g - 1 . g x = x , we h a v e b y t h e s a m e a r g u m e n t
o r Gg x c g G x g - 1 , w h i c h p r o v e s t h e p r o p o s i t i o n .
This can also be expressed in the following way.
the map ~0 : X
by qg(x) = G x .
orbit being a c o n j u g a c y class
Consider
> SG into the set of subgroupsof G, defined
G operates by inner automorphisms on SG, .an
of subgroups. By proposition .
I. 3.8 the diagram
-19 -
X r > SG
. l [
X ~ > SG
g
c o m m u t e s f o r a l l g ~ G , i . e . ~0 i s a n e q u i v a r i a n t m a p . T h e r e -
f o r e ~0 i n d u c e s f o l l o w i n g p r o p o s i t i o n 1 . 3 . 5 a m a p
N
~0: X / G > S G / G , i . e . t o e v e r y o r b i t of X t h e r e c o r r e s p o n d s
a w e l l - d e f i n e d c o n j u g a c y c l a s s of s u b g r o u p s of G , c a l l e d t h e
o r b i t - t y p e o f t h e o r b i t .
P a r t i c u l a r o r b i t - t y p e s a r e t h e c o n j u g a c y c l a s s e s {e} a n d
G . L e t x be in ~ ( X o ) of o r b i t - t y p e { e } . T h e n f o r x 6 ~ ( x ) 0
t h e r e i s one a n d o n l y one g C G s u c h t h a t gx ~ = x . B e c a u s e
= l g l x = i m p l i e s g~ lg 1 = e a n d gl = g z " glXo gzXo o r g~ o Xo
If Xo i s i n ~] (x o) of o r b i t - t y p e G , t h e n Xo i s G - i n v a r i a n t ,
a n d (Xo) - x o . T h e r e f o r e t h e f i x e d p o i n t s a r e e x a c t l y t h e
o r b i t s of o r b i t - t y p e G .
E x a m p l e 1. 3 .9 . C o n s i d e r t h e f u l l l i n e a r g r o u p G L ( n , JR),
c o n s i s t i n g of t h e r e a l q u a d r a t i c m a t r i c e s w i t h n e n t r i e s h a v i n g
a d e t e r m i n a n t d i f f e r e n t f r o m z e r o , w i t h t h e n a t u r a l o p e r a t i o n
on n~ n . T h e o r i g i n O and i t s c o m p l e m e n t n~ n - { o } a r e t h e
o r b i t s of t h i s o p e r a t i o n . T h e o r b i t - t y p e of O i s G L ( n , ]1%).
- 2 0 -
E x a m p l e 1. 3 .10. C o n s i d e r IR n w i t h t h e s t a n d a r d e u c l i d e a n
m e t r i c a n d the c o r r e s p o n d i n g o r t h o g o n a l g r o u p O ( n , JR). T h e
o r b i t s of t h e n a t u r a l o p e r a t i o n a r e t he s p h e r e s w i t h t he o r i g i n
as c e n t e r . T h e i s o t r o p y g r o u p of a po in t d i f f e r e n t f r o m the
o r i g i n i s i s o m o r p h i c to t he o r t h o g o n a l g r o u p O(n-1 , JR).
E x a m p l e 1. 3.11. L e t X d e n o t e t h e c o m p l e x p l a n e w i t h a
p o i n t a t i n f i n i t y a d j o i n e d . T h e g r o u p of t r a n s f o r m a t i o n s of t he
az +b t y p e z " " > w i t h a, b, c d 6 ~; a n d ad - bc ~ O
cz + d
o p e r a t e s on X . X is the o r b i t of a n y po in t x ~ X .
E x a m p l e 1. 3.1Z. C o n s i d e r t h e o p e r a t i o n of a g r o u p G on
i t s e l f by i n n e r a u t o m o r p h i s m s . T h e f i x p o i n t s a r e t he e l e m e n t s
of t he c e n t e r C G . We h a v e a l r e a d y c o n s i d e r e d t h e i n d u c e d
G - o p e r a t i o n on the s e t SG of s u b g r o u p s of G . T h e o r b i t of a
s u b g r o u p i s i t s c o n j u g a c y c l a s s . T h e r e f o r e t he i n v a r i a n t s u b -
g r o u p s of G a r e e x a c t l y t h e f i x p o i n t s u n d e r t h i s o p e r a t i o n .
M o r e o v e r i t f o l l o w s t h a t t h e d i f f e r e n t c o n j u g a c 7 c l a s s e s f o r m
a p a r t i t i o n of S G .
T h e e f f e c t of an e q u i v a r i a n t m a p on t h e i s o t r o p y g r o u p s
i s d e s c r i b e d by the
-21-
PROPOSITION i. 3.13. L e t X be a G - s e t , X v G' a -set,
P: G >G' a homomorphism and ~0 : X >X' a
p-equivariant m a p . T h e n p (G x) c G ~ ( x ) .
Proof: Let g ~ G , i.e. gx= x. Then x
p (g)~0(x) = ~0 (gx) = ~(x) i.e. p(g) ~ G' ' ( p ( x ) "
G ! E x e r c i s e 1. 3 . 1 4 . L e t X b e a G - s e t , X v a - s e t ,
p: G > G' "~ a h o m o m o r p h i s m a n d ~o : X / G > X ' / G ' a m a p .
S t u d y t h e c o n d i t i o n s u n d e r w h i c h ~ i s i n d u c e d by a p - e q u i v a r i a n t
m a p ~0 : X > X t i n t h e s e n s e of p r o p o s i t i o n 1 . 3 . 5 .
Exercise I. 3. 15. For a G-set X consider the map
= Tr X o ir X : PX > PX defined above. Show that
h a s t h e f o l l o w i n g p r o p e r t i e s :
a) ~ (~)) = (~ fo~.the empty set r of X
b) M c ~ (M) f o r M c X
f o r a f a m i l y ( M x ) x 6 A of M X c X .
-ZZ -
T h e r e f o r e Q i s a " K u r a t o w s k i - o p e r a t o r " on X and d e f i n e s
a t o p o l o g y on X a c c o r d i n g to the d e f i n i t i o n : M c X is c l o s e d
if and o n l y if ~ ( M ) = M . T h i s r e m a i n s t r u e i f we c o n s i d e r
an a r b i t r a r y e q u i v a l e n c e r e l a t i o n R on X (no t n e c e s s a r i l y
d e f i n e d by a g r o u p G) and the map ~= ~-I o ~r : PX x X
>PX,
w h e r e lr : X x
> X / R is the c a n o n i c a l m a p onto t h e q u o t i e n t
s e t X / R . Show t h a t m o r e g e n e r a l l y f o r an a r b i t r a r y r e l a t i o n
R on a set X the "saturation-operator " ~: PX > PX
d e f i n e d by
(M) = [ y ~ X / x R y f o r s o m e x ~ M ]
f o r M c X s a t i s f i e s t he p r o p e r t i e s 1) to 4) if a n d o n l y i f R
is a r e f l e x i v e and t r a n s i t i v e r e l a t i o n on X .
E x e r c i s e 1 . 3 . 1 6 . C o n s i d e r t he t o p o l o g y d e f i n e d in e x e r c i s e
1. 3.15 on a s e t X e q u i p p e d w i t h an e q u i v a l e n c e r e l a t i o n R .
Show t h e f o l l o w i n g p r o p e r t i e s :
1) M c X is c l o s e d i f a n d o n l y if M is a u n i o n of
e q u i v a l e n c e c l a s s e s ;
2) M c X is closed if and only if M is open.
- Z 3 -
Wha t a r e t h e c o n d i t i o n s on X / R f o r t h e t o p o l o g y in q u e s t i o n to s a t i s f y
i)
z)
3)
Exercise i. 3.17. Let
t h e s e c o n d c o u n t a b i l i t y a x i o m ,
t o be c o m p a c t ,
t o be c o n n e c t e d ?
X be a G - s e t , R a n a r b i t r a r y s e t a n d ~ : X ' > R
a map. Suppose X equipped with the topology defined in exercise I. 3.15 and
R topologized by the discrete topology. Then (~ is invariant if and only if
(~ is continuous.
I. 4 Particular G-sets.
Let X be a G-set, defined by a homomorphism 7 : G
define some particular properties an operation can have.
7 is an effective operation if 7 D E F I N I T I O N 1 .4 . 1
Ker T = {e}.
> B i j X . We
is injective, i.e.,
W e o b s e r v e t h a t K e r T = N G x , an e l e m e n t of K e r T x C X
exactly an element of G contained in every isotropygroup. If
e f f e c t i v e , t h e n t h e r e e x i s t s a f a c t o r i z a t i o n T t h r o u g h G / K e r
i Bij x
G/Ker T
a n d G / K e r T o p e r a t e s e f f e c t i v e l y on X .
Example i. 4. Z The operation [~ of a group O
has the center CG = Ker ~ as kernel.
being
T is n o t
T
by inner automorphisms
- 2 4 -
D E F I N I T I O N 1 . 4 . 3
x 6 X i m p l i e s g = e .
T h i s m e a n s t h a t a t r a n s f o r m a t i o n
T i s a f r e e o p e r a t i o n , i f Tg(X) = x f o r s o m e
T f o r g ~ e h a s n o f i x p o i n t . g
F r e e m e a n s " f r e e of f i x p o i n t s " . T h e i s o t r o p y g r o u p i s r e d u c e d t o t h e
n e u t r a l e l e m e n t : G = { e } f o r e v e r y x ~ X . X i s a l s o c a l l e d a x
" G - p r i n c i p a l s e t " . N o t e t h a t a f r e e o p e r a t i o n i s e f f e c t i v e .
E x a m p l e 1 . 4 . 4 T h e o p e r a t i o n of G on G b y l e f t - t r a n s l a t i o n s i s f r e e .
L e t H b e a s u b g r o u p of G ,
o p e r a t i o n i s f r e e .
D E F I N I T I O N 1 . 4 . 5 T
t h e r e e x i s t s a g ~ G s u c h t h a t
t h e e l e m e n t g i s u n i q u e .
o p e r a t i n g on G b y r i g h t - t r a n s l a t i o n s . T h i s
i s a t r a n s i t i v e o p e r a t i o n , i f f o r x 1 , x z ~ X
7g(Xl) = x z , s i m p l 7 t r a n s i t i v e , i f , m o r e o v e r ,
A s i m p l y t r a n s i t i v e o p e r a t i o n i s f r e e . C o n v e r s e l y , a f r e e o p e r a t i o n
i s s i m p l y t r a n s i t i v e on e a c h o r b i t . B e c a u s e i f x = g iXo ( i = 1, Z) , t h e n
-I -i -i ={e}, i.e. gl = gz" Xo = gz x = gz glXo and therefore gz gl C Gxo
The definition of a transitive operation can also be put in the following
form: there exists an element x C X such that ~(Xo) = X. X is then O
the orbit of each point x C X. This shows that the set of orbits X/G is
a point. This property allows us to define the transitivity of a G-operation
in an arbitrary category, as so~n as the notion of point is defined.
DEFINITION i. 4. 6 A G-set X is called homogeneous, if G
transitively on X.
o p e r a t e s
- 2 5 -
E x a m p l e 1 .4 . 7 T h e o r t h o g o n a l g r o u p O(n, JR) o p e r a t e s t r a n s i t i v e l y
on the un i t s p h e r e S n-1 in IR n .
M o r e g e n e r a l l y , an o p e r a t i o n of G on X d e f i n e s a t r a n s i t i v e o p e r a t i o n
on e a c h G - o r b i t .
E x a m p l e 1 .4 . 8 T h e g r o u p of h o l o m o r p h i s m s of t h e un i t d i s k in t he
c o m p l e x p l a n e o p e r a t e s t r a n s i t i v e l y .
A f u n d a m e n t a l e x a m p l e of a h o m o g e n e o u s G - s e t is o b t a i n e d in t he
f o l l o w i n g w a y . C o n s i d e r a g r o u p G and a s u b g r o u p H o p e r a t i n g on G by
r i g h t - t r a n s l a t i o n s . T h e n we c a n d e f i n e an o p e r a t i o n of G on t h e o r b i t s e t
G / H . T h e l e f t t r a n s l a t i o n L : G > G s a t i s f i e s L g ( v H ) = g v H and g
t h e r e f o r e d e f i n e s by r ( v H ) = g v H a m a p �9 : G / H > G / H . r is g g
t he d e s i r e d o p e r a t i o n , m a k i n g G / H a G - s e t , w h i c h i s e v i d e n t l y h o m o g e n e o u s .
R e m a r k . T h e i s o t r o p y g r o u p of H i s H .
We s h a l l s h o w t h a t f o r an a r b i t r a r y h o m o g e n e o u s G - s e t X t h e r e
e x i s t s a s u b g r o u p H of G and an e q u i v a l e n c e ~ : G / H > X of G - s e t s ,
w h e r e G / H is c o n s i d e r e d as a G - s e t in t he s e n s e i n d i c a t e d .
F i r s t l e t X be an a r b i t r a r y G - s e t and x ~ X . We put H = G and O x o
d e f i n e ~ : G / H > ~ ( X o ) X by r = g x o .
L E M M A 1 . 4 . 9 @
P r o o f : F o r V ~ G
( q~ o - r y ) ( g H ) = ~0 ( "vgH) =
t he e q u i v a r i a n c e of
i s e q u i v a r i a n t and i n j e c t i v e .
one has(T~o~)(gH) =v v(gx o) = Vgx o and
�9 ~ = ~D o 0 " i . e . Vgx o , and t h e r e f o r e v V V '
~ . T o s h o w t h e i n j e c t i v i t y , c o n s i d e r g l ' gz ~ G
- Z 6 -
s u c h t h a t q~ (g l H) = O ( g z H ) .
-1 a n d t h e r e f o r e gz gl E; H . B u t gl
I f X i s h o m o g e n e o u s , t h e n
We h a v e p r o v e d
This means glXo = gZXo - i
o r gz glXo = Xo
E; gz H implies gl H = gz H , q . e . d .
Q ( x o) = X a n d cp i s a n e q u i v a l e n c e .
P R O P O S I T I O N 1 . 4 . 1 0 L e t X be a h o m o g e n e o u s G - s e t a n d s e l e c t
x 0
on G / H i n d u c e d by t h e l e f t - t r a n s l a t i o n of G . T h e n t h e m a p @ : G / H
d e f i n e d by ~ ( g H ) = g x o i s a n e q u i v a l e n c e of G - s e t s .
T h e g r o u p H d e p e n d s on t h e c h o i c e of x o 6 X , b u t t h e c o n j u g a c y
c l a s s of H i s w e l l - d e f i n e d by t h e o p e r a t i o n i n v i e w of t h e t r a n s i t i v i t y .
6 X . Le___tt H be t h e i s o t r o p y g r o u p of x o a n d c o n s i d e r t h e G - o p e r a t i o n
> X
We c o n c l u d e t h i s c h a p t e r b y s o m e r e m a r k s on e f f e c t i v e a n d t r a n s i t i v e
o p e r a t i o n s on s e t s . I n v i e w of t h e p r e c e d i n g p r o p o s i t i o n , w e c a n c o n s i d e r
w i t h o u t l o s s of g e n e r a l i t y G - s e t s of t h e t y p e G / H , w h e r e H i s a s u b g r o u p
of G . T h e k e r n e l K of t h e h o m o m o r p h i s m d e f i n i n g t h e o p e r a t i o n of G
on G / H i s t h e i n t e r s e c t i o n of t h e i s o t r o p y g r o u p s , t h e r e f o r e
K = N gHg I .
g C G
K is an invariant subgroup of G
an invariant subgroup of G with
= i l i gH g l ' H f o r s o m e ~ L i n v i e w of
s i g n i f i e s L c K . T h e r e f o r e w e h a v e
c o n t a i n e d in H . C o n v e r s e l y , i f L i s
L c H , t h e n L c K , b e c a u s e
L g = g L a n d l g H = g H w h i c h
-27 -
PROPOSITION I. 4. Ii Let G be a sroup, H a subgroup and_consider
the G-operation on G/H induced by the left-translations of G. The kernel
K of the homomorphism ~: G > Bij {G/H) definin~ this operation is
the g r e a t e s t i n v a r i a n t s u b g r o u p of G c o n t a i n e d in H and c a n be d e s c r i b e d
a s
-I K = fl gHg
g~G
C O R O L L A R Y 1.4. 12 G o p e r a t e s e f f e c t i v e l y on G / H if and only if
H c o n t a i n s no i n v a r i a n t s u b g r o u p of G d i f f e r e n t f r o m { e } .
Exercise I. 4.13 Study the effect of the choice of the point x o G X
in proposition I. 4. i0.
- 2 8 -
C h a p t e r Z. G - S P A C E S
Z. 1 D e f i n i t i o n and e x a m p l e s .
D E F I N I T I O N Z, 1.1 A t o p o l o g i c a l g r o u p G is a g r o u p w h i c h is a
t o p o l o g i c a l s p a c e s u c h t h a t t he m a p s
G x G > G
( g l ' gz ) ~ g lg2
, G > G
-1
a r e c o n t i n u o u s .
D E F I N I T I O N Z. 1. Z L e t G b e a t o p o l o g i c a l g r o u p . A G - s p a c e X
is a t o p o l o g i c a l s p a c e w h i c h i s a G - s e t w i t h r e s p e c t to a m a p G x X > X .
M o r e o v e r t h i s m a p i s s u p p o s e d to be c o n t i n u o u s . T h e p a i r (G,X) i s a l s o
c a l l e d a t o p o l o g i c a l t r a n s f o r m a t i o n g r o u p .
I t i s c l e a r t h a t t he g r o u p G is a c t i n g by h o m e o m o r p h i s m s on X ,
so t h a t X i s a G - o b j e c t in t he c a t e g o r y of t o p o l o g i c a l s p a c e s . We
r e q u i r e , m o r e o v e r , the c o n t i n u i t y of t h e m a p G x X > X .
L e t G a n d G' be t o p o l o g i c a l g r o u p s .
D E F I N I T I O N Z. 1. 3 A h o r n o m o r p h i s m 9 : G > G w of t o p o l o g i c a l
g r o u p s i s a h o m o m o r p h i s m of g r o u p s , w h i c h i s c o n t i n u o u s .
L e t X be a G - s p a c e , X' G' a a - s p a c e a n d P: G > G '
h o m o m o r p h i s m .
-Z9 -
DEFINITION Z. i. 4 A 9-equivariant map ~: X > X' is a
p -equivariant map in the sense of definition I. 2.1 which is continuous.
The map r makes the following diagram commutative
G x X > X
[ [ px ] $
X' X' G' x >
~0 i s c o n t i n u o u s and t h e r e f o r e a l s o P
the u n i v e r s a l p r o p e r t y of t he p r o d u c t t o p o l o g y .
An e q u i v a l e n c e of G - s p a c e s X, X t is an e q u i v a l e n c e
of G - s e t s w h i c h i s a h o m e o m o r p h i s m .
E x a m p l e Z. 1 .5 L e t G be a t o p o l o g i c a l g r o u p .
x @, as follows immediately by
~:X >X'
T h e o p e r a t i o n of
G on G by l e f t o r r i g h t - t r a n s l a t i o n s m a k e s t he s p a c e G a G - s p a c e .
T h e o p e r a t i o n of G on G by i n n e r a u t o m o r p h i s m s a l s o m a k e s G a
G - s p a c e .
R e m a r k . L e t X be a t o p o l o g i c a l s p a c e and G the g r o u p of
h o m e o m o r p h i s m s of X . T h e d i s c r e t e t o p o l o g y on G c e r t a i n l y m a k e s
X a G - s p a c e .
L e t X be a c o m p a c t G - s p a c e . C o n s i d e r t h e g r o u p Aut X
of h o m e o m o r p h i s m s w i t h t he c o m p a c t - o p e n t o p o l o g y . I t c a n be p r o v e d
t h a t Au t X i s a t o p o l o g i c a l g r o u p , a n d t h a t t h e m a p G x X > X is
continuous .
- 3 0 -
2. Z Orbitspace.
Let G be a topological group and X a G-space. Consider the set
of orbits X/G and the canonical map w : X > X/G. The quotient x
topology on X/G is the strongest topology on X/G making w x continuous.
T h e o p e n s e t s of X / G a r e t h e s e t s h a v i n g a n o p e n s a t u r a t i o n in X .
D E F I N I T I O N Z. Z. 1 T h e o r b i t s p a c e X / G of t h e G - s p a c e X i s
t h e s e t of o r b i t s w i t h t h e q u o t i e n t t o p o l o g y .
P R O P O S I T I O N Z. Z. Z w : X : > X / G i s a n o p e n m a p . T h e X ....
t o p o l o g y on X / G i s c h a r a c t e r i z e d a s b e i n 8 t h e u n i q u e t o p o l o g y m a k i n g
t h e m a p w c o n t i n u o u s a n d o p e n . X
P r o o f : L e t M C X be o p e n . 7g (M) i s o p e n a n d t h e r e f o r e a l s o
(~rxl o Wx)(lVi ) b e i n g t h e u n i o n of a l l s e t s T ( M ) . B u t t h i s ~(M) ' g
m e a n s t h a t ~rx(M) i s o p e n by d e f i n i t i o n of t h e q u o t i e n t t o p o l o g y . T o
p r o v e t h e s e c o n d s t a t e m e n t , c o n s i d e r m o r e g e n e r a l l y a m a p ~ : X > Y
f r o m X to a s e t Y . T w o t o p o l o g i e s on Y m a k i n g b o t h ~ c o n t i n u o u s
a n d o p e n n e c e s s a r i l y c o i n c i d e . B e c a u s e i f O i s a n o p e n s e t of Y in t h e
i s o p e n i n X a n d ~ ( ~ - 1 ( O ) ) = O i s a l s o o p e n i n one t o p o l o g y , ~ - 1 ( 0 )
t h e o t h e r t o p o l o g y .
Example Z. Z. 3 L e t G be a t o p o l o g i c a l g r o u p a n d H a s u b g r o u p
of G w i t h t h e r e l a t i v e t o p o l o g y . T h e o p e r a t i o n of H on G b y r i g h t
translations makes G an H-space. The canonical map w G : G > G/H
onto the orbitspace is continuous and open.
- 3 1 -
T h e q u o t i e n t t o p o l o g y on X / G c a n a l s o b e c h a r a c t e r i z e d b y t h e
f o l l o w i n g p r o p e r t y . L e t R b e a n a r b i t r a r y t o p o l o g i c a l s p a c e . T h e
m a p x~--------> ~ o lr , s e n d i n g c o n t i n u o u s m a p s x~ : X / G - - > R i n t o
c o n t i n u o u s m a p s X > R i s i n j e c t i v e .
t h e r e f o r e n o w b e c o m p l e t e d b y
P R O P O S I T I O N Z. Z. 4 L e t G
~r : X x
s p a c e .
from X/G t o R
bijective.
PROPOSITION Z. Z. 5
a homomorphism and X, X'
T h e p r o p o s i t i o n 1. 3. 4 c a n
b e a t o p o l o g i c a l ~ r o u p , X a G - s p a c e ,
> X / G t h e c a n o n i c a l m a p o n t o t h e o r b i t s p a c e a n d R a n a r b i t r a r y
T h e c o r r e s p o n d e n c e $ ~ qt o ~ , s e n d i n 8 c o n t i n u o u s m a p s
o n t o i n v a r i a n t c o n t i n u o u s m a p s f r o m X t o R i s
map 4: X > X' induces one and only one continuous map
> X'/G' such that the loll owing diagram commutes
Let G, G' be topological groups, p: G-->G'
respectively G, G'-spaces. A p-equ/variant
~: X/G
X 0 > X'
I i ~x [ 7fX! I
X/G - ~ > X'/G'
Proof: There is only t o show t h e continuity of ~ . But t h i s is a
consequence of the continuity of ~r , o (~ in view of proposition Z. Z. 4. x
Exercise 2.2.6 Consider a G-space X with a transitive operation
of G on X S e l e c t x 6 X a n d l e t H b e t h e i s o t r o p y g r o u p of x �9 0 0
-32-
D e f i n e , as in p r o p o s i t i o n 1 .4 . 10, a m a p ~ : G / H > X . T h i s m a p i s an
e q u i v a l e n c e of G - s e t s and c o n t i n u o u s , bu t no t n e c e s s a r i l y a h o m e o m o r p h i s m .
T h e f o l l o w i n g c o u n t e r - e x a m p l e i s t a k e n f r o m B o u r b a k i . L e t ]R o p e r a t e on
TZ = IR~-/~Z by 7x(x I, x 2) = (x I + a(X), x 2 + a(8•)) where X C IR,
a:]R > ]R/Z the canonical homomorphism and Z
(x 1, x 2) ~ 3" ,
i r r a t i o n a l n u m b e r . F i x i n g (x 1, xz) 6 T z we d e f i n e
1 e m i ~ a 1 .4 . 9, o b t a i n i n g a c o n t i n u o u s i n j e c t i o n .
X = ~ (x 1, x Z) w i t h t h e r e l a t i v e t o p o l o g y . ~ : ]R
b i j e c t i o n , bu t n o t a h o m e o m o r p h i s m . B e c a u s e X i s d e n s e in
c a n n o t be h o m e o m o r p h i c to t he c o m p l e t e s p a c e ]R.
0 an
~:]R > T z as in
C o n s i d e r t he i m a g e
> X is a c o n t i n u o u s
T Z and
E x e r c i s e Z. Z. 7 L e t G be a t o p o l o g i c a l g r o u p a n d H an o p e n s u b -
g r o u p . T h e n H is c l o s e d in G . ( C o n s i d e r t h e p a r t i t i o n of G d e f i n e d
by t h e e l e m e n t s of G / H . )
E x e r c i s e Z. Z. 8 L e t G be a c o n n e c t e d t o p o l o g i c a l g r o u p a n d U a
-1 n e i g h b o r h o o d of e . T h e n e i g h b o r h o o d V = U N U h a s t he p r o p e r t i e s :
Vc U, V "I V n = V. Consider the sets ={gl .... gn/gi CV, i = l,...,n}.
The union V ~176 = UV n is a group, the group generated by V. e is an
inner point of V O~ as e ~ V cV c~ Any point of V ~176 is therefore an
inner point, the left-translations being homeomorphisms leaving V ~176
invariant. V ~176 is an open subgroup of G and therefore closed. As G
is connected, this shows V ~176 = G. This proves that G is generated by
an arbitrary neighborhood U of e.
- 3 3 -
E x e r c i s e 2 . 2 . 9 Let G be a topological group and G the connected o
component of the neutral element e C G, the identity component of G.
Show that G o is a closed invariant subgroup of G
- 3 4 -
C h a p t e r 3. G - M A N I F O L D S
This chapter introduces the fundamental notions of these lectures.
In the following chapters, we proceed to a detailed study of G-manifolds
and Lie groups.
3. i Definition and examples of Lie ~roups.
Manifold will mean a Hausdorff, but not necessarily connected
manifold.
DEFINITION 3. I. I A Li e group is a group G which is an analytic
manifold such that the maps
G x G > G G > G
(gl ' gz ) ~ glgz g ~ g-1
are analytic.
C ~ Differentiable shall always mean . If one replaces analycity
by differentiability in the definition above, it doesn't change anything;
i. e. , analycity is then automatically satisfied (Pontrjagin, [14] ,
p. 191). For a great part of the theory, we shall only make explicit use
of d i f f e r e n t i a b i l i t y .
In the d e f i n i t i o n a b o v e ,
manifold.
analytic manifold means real analytic
Replacing it by complex analytic manifold, one obtains the
notion of a complex Lie group.
- 3 5 -
T w o a r b i t r a r y c o n n e c t e d n e s s c o m p o n e n t s G 1, 0 2 of a L i e g r o u p O
a r e a n a l y t i c a l l y d i f f e o m o r p h i c . F o r gl ~ O l ' g2 6 O 2 t h e m a p
~-~-~> g 2 g l l g i s a n e x a m p l e of s u c h a d i f f e o m o r p h i s m . A l l t h e g
c o n n e c t e d n e s s c o m p o n e n t s t h e r e f o r e h a v e t h e s a m e d i m e n s i o n a n d i t
m a k e s s e n s e t o s p e a k of t h e d i m e n s i o n of a L i e g r o u p .
E x a m p l e 3. I. 2 T h e a d d i t i v e g r o u p ]R n o r (~ n . 3rn ] R n / Z n , = ;
G L ( n , IR ) - t h e g r o u p of q u a d r a t i c m a t r i c e s w i t h n r o w s a n d d e t e r m i n a n t
d i f f e r e n t f r o m z e r o .
E x a m p l e B. 1. 3 L e t G b e a L i e g r o u p a n d T G t h e t a n g e n t b u n d l e .
T h e n T G i s a L i e g r o u p . T h i s f o l l o w s f r o m t h e f a c t t h a t T i s a f u n c t o r
c o n s e r v i n g d i r e c t p r o d u c t s .
E x a m p l e S. 1 . 4 L e t G 1 a n d G 2 b e L i e g r o u p s . T h e n t h e d i r e c t
p r o d u c t G 1 x G 2 i s a L i e g r o u p .
D E F I N I T I O N 3. 1 . 5 L e t G a n d G ' b e L i e g r o u p s . A h o r n o m o r p h i s m
p : G > G of Lie groups is a homomorphism of groups which is analytic.
Remark. It is to be noted that in the literature the term homomorphism
is often reserved for analytic homomorphisms of groups such that the mEp
p: G > 9 (G) is open.
Example 3. i. 6 Let V be an n-dimensional vector space over JR.
T h e c h o i c e of a b a s e e 1 . . . . , e n of V d e f i n e s a n i s o m o r p h i s m
G L ( V ) > G L ( n , IR) of g r o u p s , p e r m i t t i n g u s t o d e f i n e a L i e g r o u p
s t r u c t u r e on t h e g r o u p of l i n e a r a u t o m o r p h i s m s G L ( V ) of V . T h i s
- 3 6 -
s t r u c t u r e i s i n d e p e n d e n t of the c h o i c e of the b a s e . B e c a u s e t w o c h o i c e s
of t he b a s e of V c o r r e s p o n d t o t w o i s o m o r p h i s m s G L ( V ) > G L ( n , JR)
w h i c h d i f f e r b y an i n n e r a u t o m o r p h i s m of G L ( n , ]1%).
E x a m p l e 3. 1. 7 L e t G b e a L i e g r o u p and TG the t a n g e n t b u n d l e
w i t h i t s L i e g r o u p s t r u c t u r e ( e x a m p l e 3. 1. 3). C o n s i d e r t he t a n g e n t s p a c e
G e of G a t t he i d e n t i t y e of G and i t s n a t u r a l i n j e c t i o n j : G > T G . e
If G e i s e q u i p p e d w i t h the L i e g r o u p s t r u c t u r e d e f i n e d b y a d d i t i o n , j i s
a h o m o m o r p h i s m of L i e g r o u p s . T h e n a t u r a l p r o j e c t i o n p : T G > G ,
a s s i g n i n g to e a c h t a n g e n t v e c t o r i t s o r i g i n , i s a l s o a h o m o m o r p h i s m of
L i e g r o u p s . T h e s e q u e n c e
P O >G ~ > TG >G > e
e
i s e x a c t . M o r e o v e r , t h e r e e x i s t s a s p l i t t i n g , t he n a t u r a l i n j e c t i o n
s : G > T G , s a t i s f y i n g p o s = 1G.
E x e r c i s e 3 . 1 . 8 L e t G be a l o c a l l y E u c l i d e a n t o p o l o g i c a l g r o u p ,
i. e. , h a v i n g a n e i g h b o r h o o d of the i d e n t i t y e h o m e o m o r p h i c to an o p e n
s u b s e t of an E u c l i d e a n s p a c e . T h e i d e n t i t y c o m p o n e n t G O of G h a s a
c o u n t a b l e b a s e . T h e r e f o r e G i s p a r a c o m p a c t .
Exercise 3.1.9
Exercise 3.1. I0
open subgroup of G.
A L i e g r o u p i s l o c a l l y c o n n e c t e d .
T h e i d e n t i t y c o m p o n e n t G o of a L i e g r o u p i s an
- 3 7 -
3 . 2 D e f i n i t i o n a n d e x a m p l e s of G - m a n i f o l d s .
D E F I N I T I O N 3. 2 . 1 L e t G b e a L i e g r o u p . A G - m a n i f o l d X i s a
d i f f e r e n t i a b l e m a n i f o l d X w h i c h i s a G - s e t w i t h r e s p e c t t o a m a p
G x X > X . M o r e o v e r t h i s m a p i s s u p p o s e d t o b e d i f f e r e n t i a b l e . T h e
p a i r (G , X) i s a l s o c a l l e d a L i e t r a n s f o r m a t i o n g r o u p .
T h e g r o u p G i s a c t i n g b y d i f f e o m o r p h i s m s on X , s o t h a t X i s a
G - o b j e c t i n t h e c a t e g o r y of d i f f e r e n t i a b l e m a n i f o l d s . M o r e o v e r t h e
d i f f e r e n t i a b i l i t y of t h e m a p G x X
!
L e t X b e a G - m a n i f o l d , X a
h o m o m o r p h i s m of L i e g r o u p s .
> X i s r e q u i r e d .
v GI G -manifold, and p : G > a
D E F I N I T I O N 3. 2. ~ A 9 - e q u / v a r i a n t m a p ~ : X > X ~ i s a
9 - e q u i v a r i a n t m a p i n t h e s e n s e of d e f i n i t i o n 1. Z. 1 w h i c h i s d i f f e r e n t i a b l e .
E x a m p l e B. 2. 3 I R - m a n i f o l d s a r e of f u n d a m e n t a l i m p o r t a n c e f o r
t h e t h e o r y of G - m a n i f o l d s . T h e y h a v e r e c e i v e d a s p e c i a l n a m e : o n e -
p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s . W e s h a l l t a k e u p t h e s t u d y of
o n e - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s in c h a p t e r 5.
E x a m p l e 3 . 2 . 4 T h e o p e r a t i o n of a L i e g r o u p G on t h e u n d e r l y i n g
m a n i f o l d b y l e f t - t r a n s l a t i o n s d e f i n e s G a s a G - m a n i f o l d . T h e o p e r a t i o n
of G on i t s e l f b y i n n e r a u t o m o r p h i s m s a l s o d e f i n e s G a s a G - m a n i f o l d .
E x a m p l e 3. Z. 5 L e t V b e a f i n i t e - d i m e n s i o n a l ] R - v e c t o r s p a c e .
GL(V) is then a Lie group. Let G be a Lie group and 7: G >GL(V)
a homomorphism. We call 7 a representation of the Lie group G in
V,
-38-
Remark.
compact G-space X the continuity of the map G x X
expressed by the continuity of the homomorphism G
As observed at the end of section 2. I, for a locally
>X canbe
> Aut X defining
the operation, if Aut X is equipped with the compact-open topology.
One would like to describe similarly the differentiability of the map
G x X--> X for a G-space X. But for this the group Aut X of diffeo-
morphisms of X should first be turned into a manifold (modeled over
a suffi'ciently general topological vectorspace), which presents serious
difficulties. Nevertheless we shall use this viewpoint for heuristical
remarks.
Example 3.2.6 Let X be a G-manifold and T the functor assigning
to each differentiable manifold its tangent bundle. Then TX is a
TG-manifold, because T conserves direct products. G being a subgroup
of TG (example 3. i. 7), TX is also a G-manifold. This justifies many
classical notations in the theory of transformation groups, which at
first sight seem abusively short.
Example 3. Z. 7 Let G and G' be Lie groups and G' a G-manifold
w i t h r e s p e c t t o an o p e r a t i o n T : G > Au t G ' . T h e n t h e s e m i - d i r e c t
p r o d u c t G" xwG d e f i n e d in e x a m p l e 1 .1 .7 i s a L i e g r o u p w i t h t he a n a l y t i c
s t r u c t u r e of t he p r o d u c t - m a n i f o l d . T h i s g e n e r a l i z e s e x a m p l e 3. 1.4~
w h i c h c o r r e s p o n d s to t he t r i v i a l o p e r a t i o n of G on G ' .
L e t V be a f i n i t e d i m e n s i o n a l ] R - v e c t o r s p a c e . T h e g r o u p of a f f i n e
m o t i o n s of V , w h i c h i s t h e s e m i - d i r e c t p r o d u c t V x G L ( V ) w i t h r e s p e c t
- 3 9 -
to t h e n a t u r a l o p e r a t i o n of GL(V)
E x a m p l e 3. 2 . 8
s e q u e n c e
on V, i s a L i e g r o u p by t h e p r e c e d i n g .
L e t G be a L i e g r o u p a n d c o n s i d e r t he e x a c t
O > G e
J >TG P > G >e
of e x a m p l e 3. 1. 7.
i n j e c t i o n of G
G e d e f i n e d by
T h e s p l i t t i n g s : G > TG d e f i n e d by the n a t u r a l
g i v e s r i s e to an o p e r a t i o n of G on t h e a d d i t i v e g r o u p
Wg = g s ( g ) / G e ( e x a m p l e 1 .1 .7 ) . T h i s r e p r e s e n t a t i o n
of G in G e p l a y s an i m p o r t a n t r o l e in t he t h e o r y of L i e g r o u p s ( a d j o i n t
r e p r e s e n t a t i o n ) . , TG is i s o m o r p h i c to t he s e m i - d i r e c t p r o d u c t C, e xTG
w i t h r e s p e c t to t h i s o p e r a t i o n r .
-40 -
C h a p t e r 4. VECTORFIEI.nS
In this chapter we begin with the detailed theory of G-manifolds and
Lie groups. The Lie algebra of a Lie group is defined and the formal
properties of this correspondence are studied.
4. i. Realfunctions.
The adjective "differentiable" shall be omitted from now on, it being
understood that all manifolds and maps are differentiable.
Let X be a manifold and denote by CX the set of real-valued functions
on X. CX is a commutative ring with identity, the operations on functions
being defined pointwise. It can also be considered as an algebra over the
reals ]R, identifying the set of constant functions on X -with IR.
Let X' be another manifold. A map ~0 : X > X' induces a map
g)*: CX' > CX defined by g)*(f')= f' o ~ for f' 6 CX'. ~0" is a ring
homomorphism respecting identities.
structure on CX and CX', then ~* is
If we consider the ]R-algebra
a homomorphism of ]R -algebras
r e s p e c t i n g i d e n t i t i e s . T h i s s h o w s t h a t the c o r r e s p o n d e n c e X ~ C X ,
~ ~ * d e f i n e s a c o n t r a v a r i a n t s C : ~ > e f r o m the c a t e g o r y
~/ of m a n i f o l d s to the c a t e g o r y i~ of c o m m u t a t i v e r i n g s w i t h i d e n t i t y ,
r e s p e c t i v e l y c o m m u t a t i v e R - a l g e b r a s w i t h i d e n t i t y .
Now l e t X be a G - m a n i f o l d . A c c o r d i n g to p r o p o s i t i o n 1 .1 .9 and the
r e m a r k 1.1.10, CX is a G O - r i n g , i . e . a r i n g on w h i c h G o p e r a t e s f r o m the
r i g h t . If T : G > Aut X is the g i v e n o p e r a t i o n , "r* : G > Aut CX s h a l l
-41-
denote the induced operation. We r e p e a t the d e f i n i t i o n : I" f = f o T g g
f o r f ~ C X .
Exercise 4.1. i. Let X and X' be manifolds, CX and CX' the corres-
p o n d i n g s e t s of r e a l - v a l u e d f u n c t i o n s . Show t h a t an a r b i t r a r y r i n g h o m o -
morphism CX' > CX is a homomorphism of ]R-algebras.
E x e r c i s e 4.1. Z. L e t the s i t u a t i o n be a s in e x e r c i s e 4. 1.1 a n d
q~i:X > X (i = 1, 2) be m a p s s u c h t h a t ~1 = ~ Z " Show t h a t t h e n ~1 = Og"
Exercise 4. i. 3. Let the situation be as in exercise 4. I. i and consider
t he m a p
f r o m m a p s X
@ ~ - . > ~
[x, x'] > [cx' , cx]
> X' to r i n g h o m o m o r p h i s m s CX' > CX d e f i n e d by
E x e r c i s e 4. 1. g s h o w s t h a t t h i s m a p is i n j e c t i v e . Show t h a t
for paracompact manifolds X, X' this map is bijective. (Hint: Try to
imitate the theory of duality for A-modules over a ring A , considering
CX as the dual space of X. The study of the bidual space will then give the
desired result. ) This result should allow on principle a cornplete algebrai-
sation of the theory of differentiable manifolds.
Exercise 4. i. 4. A manifold X is connected if and only if the ring CX
is not decomposable in a direct product of non-trivial rings.
- 4 Z -
4. 2. O p e r a t o r s and v e c t o r f i e l d s .
L e t X be a m a n i f o l d and CX the s e t of r e a l - v a l u e d f u n c t i o n s , c o n -
s i d e r e d as an ]R - v e c t o r s p a c e .
D E F I N I T I O N 4. Z. 1. An o p e r a t o r A
A : CX > C X .
on X is an ]R-linear map
Example 4. 2.2. An automorphism of CX is an operator. A vector-
f i e l d on X is an o p e r a t o r . M o r e g e n e r a l l y , a d i f f e r e n t i a l o p e r a t o r on X
is an operator.
L e t OX d e n o t e t he ] R - a l g e b r a of o p e r a t o r s on X . If X' is a n o t h e r
manifold and ~ : X > X' a diffeomorphism, then ~ induces an isomor-
p h i s m ~ : OX > OX' by the d e f i n i t i o n ~ A = @~-1 o A o ~ . T h i s
d e f i n i t i o n m e a n s t h a t t he f o l l o w i n g d i a g r a m c o m m u t e s
CX ~= ~ CX'
A I I r , I
CX < ~0 j CX'
It i s c l e a r t h a t t h e c o r r e s p o n d e n c e X--N--> O X , ~--,----> ~Oa d e f i n e s a c o v a r i a n t
f u n c t o r 0 : ~ i s o ~ > ~ i s o f r o m the c a t e g o r y of m a n i f o l d s a n d d i f f e o m o r -
p h i s m s to t h e c a t e g o r y of ] R - a l g e b r a s and a l g e b r a i s o r n o r p h i s m s .
N o w l e t X be a G - m a n i f o l d w i t h r e s p e c t to a h o m o m o r p h i s m
v : G > Aut X. Then according to proposition I. I. 9, OX is a G-object
in the category of ]R -algebras. Moreover, the invariant elements under
- 4 3 -
t h i s o p e r a t i o n f o r m a n ] R - s u b a l g e b r a of OX , a s f o l l o w s i m m e d i a t e l y .
L e t u s c o n s i d e r a n a r b i t r a r y a s s o c i a t i v e A - a l g e b r a O o v e r a r i n g 1%
w i t h i d e n t i t y . T h e n o n e c a n d e f i n e a n e w m u l t i p l i c a t i o n [ , ] : O x O > O
i n t h e f o l l o w i n g w a y :
[A I, A2] = AIA 2 - AzA I for A I, A z g O
This multiplication is bilinear and satisfies
I) [A , A ] : O for A g O
Z) [A I, [A Z, A3] ] + [Az,[A 3, AI] ] + [A 3. [A l, Az]] = o
for AI, A z, A 3 g O (Jacobian identity)
t u r n i n g t h e r e f o r e O i n t o a L i e - a l g e b r a a c c o r d i n g t o
D E F I N I T I O N 4. 2. 3. A / % - m o d u l e O o v e r a r i n g 1% w i t h a b i l i n e a r
map [ , ] : O x O----> O satisfying [A, A] = O for A e O and the Jacobian
i d e n t i t y i s a L i e z a ! g e b r a o v e r A .
D E F I N I T I O N 4. 2. 31. A h o m 0 m o r p h i s m h : O > O ' of L i e _ a l j e b r a s
O a n d O t o v e r a f i e l d A i s a / % - l i n e a r m a p s a t i s f y i n g
h[A I, AZ] = [hA1, hAg] f o r A1, A g e O .
S t a r t i n g f r o m a n a s s o c i a t i v e / % - a l g e b r a O w e h a v e a s s o c i a t e d t o O
a t% - L i e a l g e b r a . T h i s c o n s t r u c t i o n i s f u n c t D r i a l , i . e . i f h : O > O I
i s a h o m o m o r p h i s m of /% - a l g e b r a , t h e n t% i s a l s o a h o m o m o r p h i s m of t h e
a s s o c i a t e d A - L i e a l g e b r a . A p p l y i n g t h i s t o t h e ] 1 % - a l g e b r a of o p e r a t o r s
on X , w e o b t a i n
-44-
PROPOSITION 4. Z. 4. Let X be a G-manifold and OX the set of
operators on X. T h e definition (Tg) ,~(A) =Tg ~-I
oA o T for A C OX makes g
OK a G-set. This operation conserves the ]R-algebra structure on OX a___s
w e l l a s t h e a s s o c i a t e d s t r u c t u r e of a n ] R - L i e a l g e b r a . In p a r t i c u l a r , t h e
invariant elements under this operation form a ]R-algebra and _a JR-Lie
algebra respectively.
PROPOSITION 4. Z. 5. Let X be a G-manifold, X' G' a -manifold
and ~:X > X' a p-equivariant diffeomorphism with respect to a homo- . . . .
morphism p : G > G'. Then @~ : OX > OX' is a p -equivariance
w i t h r e s p e c t to t h e o p e r a t i o n s d e f i n e d i n p r o p o s i t i o n 4. 2 . 4 . M o r e o v e r ,
X'.
s e n d s G - i n v a r i a n t o p e r a t o r s on X i n t o p ( G ) - i n v a r i a n t o p e r a t o r s on
This follows from remark I. i. I0 and propositions i. Z. 9 and i. g. IZ.
We now apply this to vectorfields. Let X be a manifold and A a
vectorfield on X. Then A is a map A : CX > CX which satisfies
(i) A(f I + fz) = Af I +~ for fl' fz C CX
(ii) A(flf2) = Afl-f z + fI.AIz for fl' f2 ~ CX
(iii) A(?~) = O for X C ]R
T h e r e f o r e A C O X . In f a c t , t h e s e p r o p e r t i e s a r e c h a r a c t e r i s t i c f o r v e c t o r -
f i e l d s . T h e c o m p o s i t i o n of v e c t o r f i e l d s i n OX is n o t a v e c t o r f i e l d , b u t t h e
c o m p o s i t i o n of v e c t o r f i e l d s w i t h r e s p e c t t o t h e a s s o c i a t e d ] R - L i e a l g e b r a
s t r u c t u r e [ , ] : OX x OX ~ OX g i v e s a v e c t o r f i e l d . H e r e ( i i ) i s e s s e n t i a l .
-45-
T h u s t h e v e c t o r f i e l d s f o r m a s u b a l g e b r a of t h i s ] R - L i e a l g e b r a . L e t DX
d e n o t e t h e J R - L i e a l g e b r a of a l l v e c t o r f i e l d s on X .
If X and X' are manifolds and ~: X > X a diffeomorphism, then
the isomorphism @# : OX > OX' defined at the beginning of this section
certainly sends DX into DX' . Applying proposition 4. 2.4 we therefore
obtain
COROLLARY 4. Z. 6. Let X be a G-manifold and DX the ]R-Lie
,-i 7 g ) , ( f o r algebra of vectorfields on X. The definition ( A) = 7g oA �9 7g
A ~ DE makes DE a G-Lie algebra with respect to 7: G > Aut DE.
In p a r t i c u l a r , t h e i n v a r i a n t e l e m e n t s of DX u n d e r t h i s o p e r a t i o n f o r m a
] R - L i e a l g e b r a .
A n d p r o p o s i t i o n 4. Z. 5 g i v e s
COROLLARY 4. Z. 7. L e t X b e a G - m a n i f o l d , X I ! a G -manifold
and ~0 : X !
> X a p - e q u i v a r i a n t d i f f e o m o r p h i s m w i t h r e s p e c t to a h o m o -
morphism p: G > G' Then ~, : DX > DX' �9 is a P-equivariance with
r e s p e c t t o t h e o p e r a t i o n s d e f i n e d i n c o : r o l l a r y 4. Z. 6. M o r e o v e r , g)$ s e n d s
X' G-invariant vectorfields on X into p (G)-invariant vectorfields on .
F o r l a t e r u s e , we m a k e e x p l i c i t t h e e f f e c t of q g , .
L E M M A 4. 2 . 8 . L e t ~0: X !
> X be a diffeomorphism and ~$ : DX
, -1 t h e i n d u c e d i s o m o r p h i s m on v e c t o r f i e l d s , d e f i n e d b y cpsA = r o A o q~
> DX'
�9 L e t
f' = Ax(cp#f' ). I__f cP. : T (X) x6 X and f' C CX'. Then ( ~A)~ (x) x x
! > T (x)(X) denotes, the linear map of tangent spaces induced by r
( ~ , A ) ~ ( x ) = ~ , A . X X
then
-46 -
Proof: ((~,A) f')($(x)) = ~*((r f'))(x) = ((A~*)f')(x) by
definition of ~0,. This means ( ~0,A)~(x)f' = Ax({p*f'). The right side is
exactly the definition of (~x Ax)f' and therefore also (~A)~(x) = ~,xAX.
4. 3. The Lie algebra of a Lie group.
L e t G be a L i e g r o u p � 9
D E F I N I T I O N 4. 3.1. T h e L i e a l s e b r a L G of G i s t h e IR - L i e a l g e b r a
of i n v a r i a n t v e c t o r f i e l d s u n d e r t h e o p e r a t i o n of G on G b y l e f t - t r a n s l a t i o n s .
E x p l i c i t l y s t a t e d , t h i s m e a n s t h a t t% 6 L G i f a n d o n l y i f ( L g ) . A = A
f o r a l l g E G . L G i s a L i e a l g e b r a b y c o r o l l a r y 4. Z . 6 . T h e l e t t e r L s h a l l
r e m i n d u s of l e f t i n v a r i a n t a s w e l l a s t h e f o u n d e r of t h e t h e o r y , S o p h u s L i e .
T h e f o l l o w i n g l e m m a i n s u r e s t h e e x i s t e n c e of m a n y l e f t i n v a r i a n t
vectorfields on a Lie group.
_ t h e L E M M A 4. 3. Z. L e t G be a L i e g r o u p , L G i t s L i e a l g e b r a , G e
t a n g e n t s p a c e of G a t t h e i d e n t i t y e a n d A ~ G e e
a n d on l F one A C L G s u c h t h a t A = A e e
P r o o f : I f A e x i s t s , t h e n ~ = ( L g ) , A f o r g C (3
Ag = ( ( L g ) . A)g . In v i e w of l e m m a 4. Z. 8 t h i s m e a n s
T h e n t h e r e e x i s t s one
a n d i n p a r t i c u l a r
(1) Ag = ( L g ) . e A e "
T h i s n e c e s s a r y c o n d i t i o n f o r X s h o w s t h e u n i q u e n e s s . We n o w d e f i n e A
= 1G b y t h i s f o r m u l a . A s L e , w e c e r t a i n l y h a v e A = A . T h e l e f t e e
invariance of A is seen from
- 4 7 -
((Lg),A)g~ = ( L g ) , A = ( L g ) , ( L ) , e A
= ( L g ) , A = A . e e g'~
T h e r e r e m a i n s to s h o w t h a t t he f a m i l y ( A g ) g ~ G is a v e c t o r f i e l d (i. e.
~ w
a d i f f e r e n t i a b l e v e c t o r f i e l d ) , w h i c h m e a n s t h a t A(CG) c C G . Le t f ~ C G .
By lemma 4.2.8
{ ( L g ) , A ) g f = Ae(L$f)g
and t h e r e f o r e
{Af)(g) = A e ( L g f ) .
Let N: I > G, I an interval of IR o-ontaining
d Nt/t = A Then G wi t h ~-~ = o e
O , a d i f f e r e n t i a b l e c u r v e i n
A e ( L g f) = ~ L f) ( ~ ) t = o = ~'[ f( g ~t) t - 0
w h i c h s h o w s ~ f 6 C G . |
T h e c o r r e s p o n d e n c e A "~> A e
of t he l e m m a d e f i n e s a b i j e c t i v e m a p
~ : G > L G e
w h i c h i s s e e n to be an i s o m o r p h i s m of I R - v e c t o r s p a c e s by (1). We h a v e
p r o v e d
- 4 8 -
G e
T H E O R E M 4. 3. 3. L e t G be a L i e g r o u p , L G i t s L i e a l g e b r a a n d
t h e t a n g e n t s p a c e of G a t t h e i d e n t i t y e . T h e f o r m u l a
( ~ ( A e ) ) g = ( L g ) . e A e f o r g C G , A e C G e
defines a map ~: G ----> 113, which is an isomorphism of ]R-vectorspaces. e
T h i s m a p ~ a l l o w s t r a n s p o r t i n g t h e s t r u c t u r e of ] R - L i e a l g e b r a f r o m
L G to G e
In t h i s s e n s e , G i s o f t e n r e f e r r e d to a s t h e L i e a l g e b r a of G. e
C O R O L L A R Y 4. 3 . 4 . L e t G be a L i e g r o u p of d i m e n s i o n n . T h e
L i e a l g e b r a L G is a L i e a l g e b r a of d i m e n s i o n n .
C o n s i d e r t h e m a p ( L g ) . : G > G , w h i c h i s a n i s o m o r p h i s m f o r e e g
all g6; G. More generally, the maps P(gl' gz ) = (Lgz)*e(Lgl):1~e : Gg I >Ggz
h a v e t h e p r o p e r t i e s :
1) P ( g z ' g3 ) p ( g l ' gz ) = P ( g l ' g3 ) f o r g l ' gz ' g3 6; G
Z) P(g, g) = IGg for g 6; G
(x) > T (x) DEFINITION 4. 3.5. Let X be a manifold and P(gz' gl ) : T 1 gz
a n J R - l i n e a r m a p f o r a l l ( g l ' gz ) 6; X x X , s a t i s f y i n g 1) a n d Z). T h e n X
i s c a l l e d a p a r a l l e l i z a b l e m a n i f o l d .
The maps P(gz' gl ) are then necessarily isomorphisms and it makes
s e n s e t o s p e a k of t h e d i m e n s i o n of X.
L e t e be a f i x e d p o i n t of X a n d A i (i = 1, �9 �9 �9 , n , n , - d i m X) a b a s e e
of t h e v e c t o r s p a c e T X . T h e n P ( e , g) A. = A. d e f i n e s v e c t o r f i e l d s e 1 e lg
A. ( i = 1, �9 . . , n) on X such tha t thex~ec to rs A. ( i = 1, . - - , n) f o r m a base 1 lg
-49 -
of T X for all g C G. g
COROLLARY 4. 3. 6. The manifold of a Lie group G is parallelizable.
Example 4. 3. 7. Consider IR with its additive Lie group structure.
Then LIR ~ IR as vectorspace, because the tangent space of IR at O is
]R . There is only one possible Lie algebra structure on 11% , defined by
= Ofor c m.
By the same argument, L 11 ~ = IR for the additive group ~Ir= IR/~.
Now let V be n-dimensional IR-vectorspace and G = GL(V). We
first remark that considering GL(V) c s = algebra of ]R-linear
endomorphiams of V, the tangent space G is identified to s (V) for all g
g G G. The multiplication in GL(V) is the restriction of the bilinear
map s (V) x s (V) > s defining the multiplication in s This
shows that (Lg)~ A,~ = gay for g ~ GL(V), Ay C Gy identified to s
Y We s h o w n o w
P R O P O S I T I O N 4. 3 . 8 . A f t e r t h e c a n o n i c a l i d e n t i f i c a t i o n of L { G L ( V ) )
w i t h t h e t a n g e n t s p a c e a t t h e i d e n t i t y , w e h a v e L ( G L ( V ) ) = s (V) a s L i e
a l g e b r a s , w h e r e on s w e c o n s i d e r t h e L i e a l g e b r a s t r u c t u r e a s s o c i a t e d
in t h e s e n s e of s e c t i o n 4. 2 t o t h e n a t u r a l a l g e b r a s t r u c t u r e .
P r o o f : L e t A 1, A 2 C L ( G L ( V } } . W e u s e t h e f o r m u l a
[AI'Az] g = Z -
which is valid for the global chart given by the embedding GL(V)c s
-50 -
In view of A i = gA i g e
which shows
w e h a v e
I~ Aigl(g) Ajg = AjgAig
[A I, A z] g = AlgAZg - AZgAlg.
B u t t h e r i g h t s i d e i s j u s t t h e c o m m u t a t o r [ A l g , A z ] in s F o r g
g = e t h i s g i v e s t h e d e s i r e d r e s u l t .
We have defined the Lie algebra LG of a Lie group G by considera-
tion of the operation of G on G by left translation. Doing the same for
the right translations, we obtain another Lie algebra RG. Explicitly:
RG i s the Lie algebra of the right invariant vectorfields. As in theorem
4. 3. 3, we can define an isomorphism O > RG of ]R-vectorspaces, e
obtaining therefore an isomorphism LG ~ RG of ]R-vectorspaces. We
shall see in section 4. 6 that there is also a natural isomorphism LG ~ RG
of the ]R-Lie algebra structure.
Exercise 4. 3.9. Let G be a Lie group, CG the ]R-vectorspace
of real-valued functions on G , DG the ]R-Lie algebra of all vectorfields
on G and LG the Lie algebra of G. Show that D G = CG | LG.
4. 4. E f f e c t of m a p s on o p e r a t o r s a n d v e c t o r f i e l d s .
I n s e c t i o n 4. 2. w e h a v e s e e n t h e e f f e c t of d i f f e o m o r p h i s m s on o p e r a t o r s .
We w a n t to s t u d y n o w t h e e f f e c t of a r b i t r a r y ( i . e . d i f f e r e n t i a b l e ) m a p s .
-51 -
map
v A ! ! Let X, X be manifolds and A, operators on X, X respectively.
!
DEFINITION 4. 4.1. A and A are ~0-related with respect to a
&0: X > X' , if the following diagram commutes.
C X
t AI
I CX (0"
CX'
CX'
If ~ is a diffeomorphism, A and ~0~A are @-related operators.
A ! But in the general case, A does neither determine an such that A and
A' A' are ~0-related, nor is unique, if it exists.
I
LEMMA 4. 4. g. Let ~0: X > X be a map.
(i) mIf A i and A v.1 (i = i, 2) are ~-related operators on
X and X' respectively, then the following operators
are ~-related:
! I
A I + A 2 and A I + A 2 ,
I !
AIA g and AIA 2 ,
[A I, A2] and [ AVl , A'Z]
Proof:
(ii) If A and A' are ~0-related operators on X and X'
res]~ectively, then for k C IR ~IA and XA' are
~0 -related.
(i) Let f' ~ CX'. Then
, �9 . , _ , ~0~(AV2f ') (D*((A I + A'z)f') = @$(A' I f' + A'zf') = ~ (All) +
= AI(~f') + AZ(~*f') = (A I + AZ)(~f'),
-SZ-
! !
showing that A I + A z and A 1 + A z are W-related. The ~-relatedness
' ' is seen by comparing the diagrams serving to define of AIA g and AIA 2
@-relatedness, and the third assertion is a consequence of this and (ii).
(ii) ~*((kA')f') = (D *(k(A f )) = @ k. ~*(A'f')
= k. A(~0*f') = ( XA)(@*f') , q. e. d.
T h e l e m m a a p p l i e s in p a r t i c u l a r to v e c t o r f i e l d s . F o r t h a t we m a k e
explicit the notion of (~-relatedness in
PROPOSITION 4.4. 3. Let X, X' be manifolds, @ : X > X' a
map and A, A' vectorfields on X, X' respectively. Then A and A'
~ x A x A~( f o r e v e r y x g X . a r e ~ - r e l a t e d if and o n l y if = x)
Proof: Let f' g CX' Then
(4. Ax)f' = Ax(~f') = (A(~f'))(x) x
by d e f i n i t i o n of ~)~ . On the o t h e r h a n d x
A ! @(x)f' = (A'f')(@(x)) = (@*(A'f'))(x)
C o m p a r i s o n p r o v e s the l e m m a .
4. 5. T h e f u n c t o r L.
We h a v e d e f i n e d the L i e a l g e b r a LG f o r a n y L i e g r o u p G. We w a n t
to e x t e n d t h i s c o r r e s p o n d e n c e to a f u n c t o r f r o m L i e g r o u p s to L ie a l g e b r a s .
L E M M A 4 . 5 . i. L e t G G' be L i e g r o u p s P : G > G ' , , a h o m o -
m o r p h i s m of L i e g r o u p s and A ~ L G . T h e n t h e r e e x i s t s one and o n l y
A' a r e O - r e l a t e d . one C LG' s u c h t h a t A and A'
-53-
Proof: Suppose A'
sition 4. 4. 3 we obtain
(i)
e x i s t s w i t h t he d e s i r e d p r o p e r t i e s . By p r o p o -
w h e r e e , e is t he i d e n t i t y of G, G'
is on ly one A' 6 IX]' s u c h t h a t ~', e
N o w w e d e f i n e c o n v e r s e l y A' a s t he u n i q u e e l e m e n t of LG '
A' ,eAe e ) = p
respectively. By lemma 4. 3. Z there
A' = , . This proves uniqueness. e
s ati s lying
!
p , g A g = 9 , g ( L g ) , e A e = ( L g ( g ) ) * e ' 9 * e Ae = A p ( g )
w h i c h i s the d e s i r e d r e s u l t in v i e w of p r o p o s i t i o n 4. 4. 3.
In t h e p r o o f of t he l e m m a 4. 5 .1 , w e u s e d on ly t he r e s t r i c t i o n of
to a neighborhood of e C G.
notion.
DEFINITION 4. 5. Z.
neighborhood of e ~ G.
T U is a differerLtiable m a p
It i s u s e f u l to i n t r o d u c e a c o r r e s p o n d i n g
L e t G , G' be L i e g r o u p s and U
A l o c a l h o m o m o r p h i s m G > G'
p : U > G' which satisfies
an open
defined on
p(g lgz ) = p ( g l ) p ( g z ) f o r a l l g l ' gz C U s u c h t h a t g lg z e U .
T h e r e s t r i c t i o n of a h o m o m o r p h i s m p : G > G' to an o p e n
n e i g h b o r h o o d of e 6 G i s a l o c a l h o m o m o r p h i s m G ~ > G ' . If we i d e n t i f y
a m a p w i t h i t s r e s t r i c t i o n s to o p e n s u b s e t s of t h e d o m a i n , we c a n c o m p o s e
l o c a l h o m o m o r p h i s m s , o b t a i n i n g t h u s the c a t e g o r y of L i e g r o u p s a n d
p o Lg - Lp(g) o p i m p l i e s
(1) T h e r e r e m a i n s to s h o w t h a t A a n d A' �9 a r e ~ - r e l a t e d . N o w
- 5 4 -
l o c a l h o m o m o r p h i s m s .
i s o m o r p h i s m . E x p l i c i t l y s t a t e d we h a v e
D E F I N I T I O N 4. 5. 3. T w o L i e g r o u p s , G and G' ,
i s o m o r p h i c if and on ly i f t h e r e e x i s t s o p e n n e i g h b o r h o o d s
l l U I and a d i f f e o m o r p h i s m p: U > U w i t h i n v e r s e p :
b o t h P and P' a r e l o c a l h o m o m o r p h i s m s .
T H E O R E M 4. 5 . 4 . L e t G, G' be L i e g r o u p s ,
h o o d of e in G and P: U ~ G'
A n e q u i v a l e n c e in t h i s c a t e g o r y is c a l l e d a l o c a l
a r e l o c a l l y
U, U t Of e, e'
> U s u c h t h a t
U a n open nei~hbor-
a local homomorphism. The formula
(L(p)A)e, = p~ A e e
defines a homomorphism of Lie algebras
following diagram is commutative,
f o r A 6 LG
L ( p ) : LG > LG ' . T h e
P e t
G > G , e e
I I
LG L(P ) > LG'
w h e r e ~ d e n o t e s the i s o m o r p h i s m of t h e o r e m 4. 3. 3. M o r e o v e r f o r
!
A C LG the v e c t o r f i e l d s A / U and L( p)A ~ LG a r e p - r e l a t e d .
P r o o f : In t he p r o o f of l e m m a 4. 5.1 a l l w a s s h o w n e x c e p t t h e f a c t
t h a t L ( p ) is a h o m o m o r p h i s m . T h i s is a c o n s e q u e n c e of l e m m a 4. 4. Z.
We o b s e r v e t h a t a h o m o m o r p h i s m p : G > G ' d e f i n e s in the
I s a m e w a y a h o m o m o r p h i s m R(p ) : RG > RG o f t h e L ie a l g e b r a s of
right invariant vectorfields.
-55-
Complement to 4. 5.4. If P : G > G' is an isomorphism, then
L(p ) = P~/LG , where p ~ : DG �9 > DG' is the map defined in section 4. Z.
P r o o f : We h a v e to s h o w t h e c o m m u t a t i v i t y of t h e d i a g r a m
LG L(P) ; LG'
N A p~'
DG > DG
L e t A 6 L G . T h e n on one h a n d
(L(p)A)p (g)
and on the other hand
p , A e = p , ( L g ) , e A e = (Lp(g))~e e
(P~A)p(g) : p. Ag g
by lemrna 4. Z. 8. This shows the desired property.
One cannot define U(p) in general by P~, because this map only
makes sense for a diffeomorphism p .
THEOREM 4. 5.5. Let ~ be the cate~or}r of Lie groups and local
homgmorphisms of Lie ~roups, s
homomorphisms of Lie algebras.
t h e c a t e g o r y of ] R - L i e a l g e b r a s a n d
T h e c o r r e s p o n d e n c e G ~ - ~ > L G ,
9 ~ " " > L ( p ) d e f i n e s a c o v a r i a n t f u n c t o r L : s > s
T h i s i s c l e a r by t h e o r e m 4 . 5 . 4 .
We c a n a l s o c o n s i d e r t h e f u n c t o r g i v e n by G " ' " > G , P ~ P e
The commutativity of the diagram in theorem 4. 5.4 expresses that
is a natural transformation of this functor into L, in fact a natural
e
M
e q u i v a l e n c e .
-56-
COROLLARY 4.5.6. The Lie algebras of locally isomorphic
groups are isomorphic.
Proof: L sends equivalences in ~ into equivalences in =s
We apply this to the natural injection G o ~ G of the connected
component of the identity G o into G , which is a local isomorphism.
This shows LG o ~ LG. The Lie algebra is therefore a property of
of an arbitrary neighborhood of the identity.
>It= IR/Z
Therefore LIR ~ L~r , what we already know.
p: "][" > ]R is a homomorphism, then p = 0.
being compact, p (,j[n) is contained in a closed interval
"IF with p(t) # 0. Then there exists a positive integer
G o , in fact,
Example 4. 5. 7. The canonical homomorphism ]R
is a local isomorphism.
LEMIVIA 4.5.8. If
Proof: "Jr
I. Suppose t C
n such that nP(t) ~ l;which is a contradiction.
This proves that there is no homomorphism
the identity isomorphism LT = LIR.
isomorphism has this property.
Example 4.5.9.
and T : G > GL(V)
"ff > IR i n d u c i n g
B u t of c o u r s e t h e n a t u r a l l o c a l
L e t V be a n ] R - v e c t o r s p a c e of f i n i t e d i m e n s i o n ,
a r e p r e s e n t a t i o n of t h e L i e g r o u p G i n V. We
p r o v e d in P r o p o s i t i o n 4. 3. 8 t h a t s (V) i s t h e L i e a l g e b r a of G L ( V ) .
T h e m a p T c a n be s e e n to be d i f f e r e n t i a b l e , a n d i n d u c e s t h e r e f o r e a
homomorphism L(T) : LG > s (V).
- 5 7 -
D E F I N I T I O N 4. 5 . 1 0 .
t h e A - L i e a l g e b r a of A - e n d o m o r p h i s m s of V.
a A - L i e a l g e b r a O i n V i s a h o m o m o r p h i s m
t h e n c a l l e d a n O - m o d u l e w i t h r e s p e c t t o v .
L e t A b e a r i n g , V a A - m o d u l e a n d s
A r e p r e s e n t a t i o n of
: 0 ~> s V is
F o l l o w i n g e x a m p l e 4. 5 . 9 , a r e p r e s e n t a t i o n of a L i e g r o u p G i n
a f i n i t e - d i m e n s i o n a l ] R - v e c t o r s p a c e V d e f i n e s a r e p r e s e n t a t i o n of t h e
L i e a l g e b r a L G in V.
W e n o w c o n s i d e r t h e h o m o m o r p h i s m d e t : G L ( V ) > ]I% $ i n t o t h e
r n u l t i p l i c a t i v e g r o u p ]R $ of t h e r e a l s . T h e L i e a l g e b r a of ]R $ i s JR.
We p r o v e
P R O P O S I T I O N 4. 5 .11. T h e h o m o m o r p h i s m s > IR of L i e
a l g e b r a s i n d u c e d b y t h e h o m o m o r p h i s m G L ( V ) > ]R $ i s t h e t r a c e
m a p .
P r o o f : L e t A ~ s (V) a n d a t a c u r v e in G L ( V ) w i t h a ~ = e ,
= A. Then 0
det, A = d {det at}/t e dt = 0 "
Now for any non-degenerated n-form co on V (n = dim V) and any
n - t u p l e of v e c t o r s Vl , ' ' - , Vn of V w e h a v e
~0(Vl, " ' ' ' V n ) " d e t a t = r t v l , ' ' ' , a t v )n
and therefore
-58-
~(Vl, - . . , Vn) �9 d e t , e d
A = d-t-{U~(0~tVl' " ' " 'atVn)}/t=O
= Z ~ ( a t V l ' " ' ' a t V i - l ' ~ tv i ' a tV i+ l ' ' ' ~ t V n ) f / = O i
= ~, ~(Vl,''',Av i, --., v n)
i
= ~ ( V l , ' ' ' , Vn) - %r A
s h o w i n g e
4 . 5 . 4 .
d e t ~ A = t r A . T h i s i s t h e d e s i r e d r e s u l t i n v i e w o f t h e o r e m
C O R O L L A R Y
P r o o f :
w e h a v e f o r
4. 5.1Z. t r ( A B ) = t r ( B A ) f o r A, B C s .
t r : s > IR b e i n g a h o m o m o r p h i s m of L i e a l g e b r a s ,
A, B C ~(V)
t r ( A B - BA) = t r [ A , B ] = [ t r A, t r B] = O ,
t h e l a t t e r b r a c k e t b e i n g the t r i v i a l one i n 1R .
Example 4.5. 13. Let G be a Lie group and
0 - - - -> G > TG -- - -> G > e e
the sequence of example 3. I. 7. It induces a sequence of Lie algebra
h o m o m o r p h i s m s
O ~ L(Ge) > L ( T G ) ~ LG > O
H e r e LG e - ' G e ( s e e e x a m p l e 4. 6. 4 b e l o w ) . S e e 7 . 5 . 6 f o r t h e e x a c t n e s s
of t h i s s e q u e n c e . T h e i n c l u s i o n G ~ TG i n d u c e s a h o m o m o r p h i s m
- 5 9 -
4. 6. A p p l i c a t i o n s of t he f u n c t o r a l i t y of L.
4. 6 .1 . T h e L i e a l g e b r a of a p r o d u c t g r o u p . L e t G 1, G 2 be Lie
g r o u p s a n d G 1 x G Z t he p r o d u c t g r o u p . T h e c a n o n i c a l p r o j e c t i o n s
Pi : G1 x G 2 > G ( i = 1, 2 ) a r e h o m o m o r p h i s m s of L i e g r o u p s a n d i n d u c e l
L i e a l g e b r a h o m o m o r p h i s m s L(Pi ) : L (G 1 x GZ) > LG.I " L e t e l , e 2 be
t h e i d e n t i t i e s of G 1, G 2 . By t h e o r e m 4 . 5 . 4 , w e h a v e t h e c o m m u t a t i v e
diagram (expressing the naturalit 7 of ~ )
1t
X
G 2
t
LG 2
t h e v e r t i c a l a r r o w s b e i n g i s o m o r p h i s m s . T h e I R - l i n e a r i s o m o r p h i s m
~-- G 1 x (G 1 x G 2 ) e l ' ez e l G 2 e 2
i m p l i e s t h e r e f o r e t h e i s o m o r p h i s m s of I R - v e c t o r s p a c e s
L(G 1 x G2) = LG 1 x LG 2 .
If q i : LGI x L G 2 > LG. (i = I, 2) denotes the canonical projection, 1
this isomorphism is given b y the commutative diagram
-60 -
L(G 1 (%
x G z) > LX31 x LG z
LG 1 LG 2
We w a n t to t r a n s p o r t t he L i e a l g e b r a s t r u c t u r e of
LG 1 x LG z. For A~ L(G 1
with A.I = L(Pi)A (i = I, 2).
a(A') = (A_[, A'Z) with A~: =
a [ A , A ' ] =
x GZ) we h a v e
S i m i l a r l y f o r A' C L(G 1
L (P i )A ' (i = 1, Z). T h e n
(L(PI)[A, A'] , L(pz)[A, A'] )
L(G 1 x G z) to
a(A) = (L(PI)A, L(pz)A) = (A I, A Z)
x G 2) we have
= ( [ A 1, A:] , [ A z , A~] )
as L(p i) a r e L i e a l g e b r a h o m o m o r p h i s m s . We d e f i n e